Section 14.5 Chain Rule

Suppose that \(z = f(x, y)\) is a differentiable function of \(x\) and \(y\), where \(x = g(t)\) and \(y = h(t)\) are both differentiable functions of \(t\). Then \(z\) is a differentiable function of \(t\) and \[ \frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} \]

EXAMPLE 1 If \(z = x^2y + 3xy^4\), where \(x = \sin 2t\) and \(y = \cos t\), find \(dz/dt\) when \(t = 0\).

EXAMPLE 2 The pressure P (in kilopascals), volume V (in liters), and temperature T (in kelvins) of a mole of an ideal gas are related by the equation \(PV = 8.31T\). Find the rate at which the pressure is changing when the temperature is 300 K and increasing at a rate of 0.1 K/s and the volume is 100 L and increasing at a rate of 0.2 L/s.

Suppose that \(z = f(x, y)\) is a differentiable function of \(x\) and \(y\), where \(x = g(s, t)\) and \(y = h(s, t)\) are differentiable functions of \(s\) and \(t\). Then \[ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \] \[ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \]

EXAMPLE 3 If \(z = e^x \sin y\), where \(x = st^2\) and \(y = s^2t\), find \(\partial z / \partial s\) and \(\partial z / \partial t\).

To remember the Chain Rule, it’s helpful to draw the tree diagram.

We draw branches from the dependent variable \(z\) to the intermediate variables \(x\) and \(y\) to indicate that \(z\) is a function of \(x\) and \(y\). Then we draw branches from \(x\) and \(y\) to the independent variables \(s\) and \(t\). On each branch we write the corresponding partial derivative. To find \(\partial z / \partial s\), we find the product of the partial derivatives along each path from \(z\) to \(s\) and then add these products: \[ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \] Similarly, we find \(\partial z / \partial t\) by using the paths from \(z\) to \(t\).

Now we consider the general situation in which a dependent variable \(u\) is a function of \(n\) intermediate variables \(x_1, ..., x_n\), each of which is, in turn, a function of \(m\) independent variables \(t_1, ..., t_m\). Notice that there are \(n\) terms, one for each intermediate variable. The proof is similar to that of Case 1.

EXAMPLE 4 Write out the Chain Rule for the case where \(w = f(x, y, z, t)\) and \(x = x(u, v), y = y(u, v), z = z(u, v)\), and \(t = t(u, v)\).

EXAMPLE 5 If \(u = x^4y + y^2z^3\), where \(x = rse^t, y = rs^2e^{-t}\), and \(z = r^2s \sin t\), find the value of \(\partial u / \partial s\) when \(r = 2, s = 1, t = 0\).

If \(g(s, t) = f(s^2 - t^2, t^2 - s^2)\) and \(f\) is differentiable, show that \(g\) satisfies the equation \[ t \frac{\partial g}{\partial s} + s \frac{\partial g}{\partial t} = 0 \]

If \(z = f(x, y)\) has continuous second-order partial derivatives and \(x = r^2 + s^2\) and \(y = 2rs\), find (a) \(\partial z / \partial r\) and (b) \(\partial^2 z / \partial r^2\).

We suppose that an equation of the form \(F(x, y) = 0\) defines \(y\) implicitly as a differentiable function of \(x\), that is, \(y = f(x)\), where \(F(x, f(x)) = 0\) for all \(x\) in the domain of \(f\). If \(F\) is differentiable, we can apply Case 1 of the Chain Rule to differentiate both sides of the equation \(F(x, y) = 0\) with respect to \(x\). Since both \(x\) and \(y\) are functions of \(x\), we obtain \[ \frac{\partial F}{\partial x} \frac{dx}{dx} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0 \] But \(dx/dx = 1\), so if \(\partial F / \partial y \ne 0\) we solve for \(dy/dx\) and obtain \[ \frac{dy}{dx} = - \frac{\partial F / \partial x}{\partial F / \partial y} = - \frac{F_x}{F_y} \]

Note

To derive this equation we assumed that \(F(x, y) = 0\) defines \(y\) implicitly as a function of \(x\). The Implicit Function Theorem, proved in advanced calculus, gives conditions under which this assumption is valid: it states that if \(F\) is defined on a disk containing \((a, b)\), where \(F(a, b) = 0, F_y(a, b) \ne 0\), and \(F_x\) and \(F_y\) are continuous on the disk, then the equation \(F(x, y) = 0\) defines \(y\) as a function of \(x\) near the point \((a, b)\) and the derivative of this function is given by Equation 6.

EXAMPLE 8 Find \(y'\) if \(x^3 + y^3 = 6xy\).

Now we suppose that \(z\) is given implicitly as a function \(z = f(x, y)\) by an equation of the form \(F(x, y, z) = 0\). This means that \(F(x, y, f(x, y)) = 0\) for all \((x, y)\) in the domain of \(f\). If \(F\) and \(f\) are differentiable, then we can use the Chain Rule to differentiate the equation \(F(x, y, z) = 0\) as follows: \[ \frac{\partial F}{\partial x} \frac{\partial x}{\partial x} + \frac{\partial F}{\partial y} \frac{\partial y}{\partial x} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial x} = 0 \] But \[ \frac{\partial}{\partial x}(x) = 1 \quad \text{and} \quad \frac{\partial}{\partial x}(y) = 0 \] so this equation becomes \[ \frac{\partial F}{\partial x} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial x} = 0 \] If \(\partial F / \partial z \ne 0\), we solve for \(\partial z / \partial x\) and obtain the first formula in Equations 7. The formula for \(\partial z / \partial y\) is obtained in a similar manner. \[ \frac{\partial z}{\partial x} = - \frac{\partial F / \partial x}{\partial F / \partial z} \quad \frac{\partial z}{\partial y} = - \frac{\partial F / \partial y}{\partial F / \partial z} \]

Again, a version of the Implicit Function Theorem stipulates conditions under which our assumption is valid: if \(F\) is defined within a sphere containing \((a, b, c)\), where \(F(a, b, c) = 0, F_z(a, b, c) \ne 0\), and \(F_x, F_y\), and \(F_z\) are continuous inside the sphere, then the equation \(F(x, y, z) = 0\) defines \(z\) as a function of \(x\) and \(y\) near the point \((a, b, c)\) and this function is differentiable, with partial derivatives given by (7).

EXAMPLE 9 Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) if \(x^3 + y^3 + z^3 + 6xyz = 1\).

Use chain rule to find \(\frac{dz}{dt}\) or \(\frac{dw}{dt}\).

\(z = x^2y^3 - x^2y\), \(x = t^2 + 1\), \(y = t^2 - 1\)

Use chain rule to find \(\frac{dz}{dt}\) or \(\frac{dw}{dt}\).

\(z = \frac{x-y}{x+2y}\), \(x = e^t\), \(y = e^{-2t}\)

Use chain rule to find \(\frac{dz}{dt}\) or \(\frac{dw}{dt}\).

\(z = \sin x \cos y\), \(x = \sqrt{t}\), \(y = 1/t\)

Use chain rule to find \(\frac{dz}{dt}\) or \(\frac{dw}{dt}\).

\(z = \sqrt{1 + xy}\), \(x = \tan t\), \(y = \arctan t\)

Use chain rule to find \(\frac{dz}{dt}\) or \(\frac{dw}{dt}\).

\(w = xe^{y/z}\), \(x = t^2\), \(y = 1-t\), \(z = 1 + 2t\)

Use chain rule to find \(\frac{dz}{dt}\) or \(\frac{dw}{dt}\).

\(w = \ln \sqrt{x^2 + y^2 + z^2}\), \(x = \sin t\), \(y = \cos t\), \(z = \tan t\)

Use chain rule to find \(\frac{ \partial z}{ \partial s}\) or \(\frac{ \partial z}{ \partial t}\).

\(z = (x - y)^5\), \(x = s^2t\), \(y = st^2\)

Use chain rule to find \(\frac{ \partial z}{ \partial s}\) or \(\frac{ \partial z}{ \partial t}\).

\(z = \tan^{-1}(x^2 + y^2)\), \(x = s \ln t\), \(y = te^s\)

Use chain rule to find \(\frac{ \partial z}{ \partial s}\) or \(\frac{ \partial z}{ \partial t}\).

\(z = \ln(3x + 2y)\), \(x = s \sin t\), \(y = t \cos s\)

Use chain rule to find \(\frac{ \partial z}{ \partial s}\) or \(\frac{ \partial z}{ \partial t}\).

\(z = \sqrt{x}e^{xy}\), \(x = 1 + st\), \(y = s^2 - t^2\)

Use chain rule to find \(\frac{ \partial z}{ \partial s}\) or \(\frac{ \partial z}{ \partial t}\).

\(z = e^r \cos \theta\), \(r = st\), \(\theta = \sqrt{s^2 + t^2}\)

Use chain rule to find \(\frac{ \partial z}{ \partial s}\) or \(\frac{ \partial z}{ \partial t}\).

\(z = \tan(u/v)\), \(u = 2s + 3t\), \(v = 3s - 2t\)

Let \(p(t) = f(g(t), h(t))\), where \(f\) is differentiable, \(g(2) = 4\), \(g'(2) = -3\), \(h(2) = 5\), \(h'(2) = 6\), \(f_x(4, 5) = 2\), \(f_y(4, 5) = 8\). Find \(p'(2)\).

Let \(R(s, t) = G(u(s, t), v(s, t))\), where \(G\), \(u\), and \(v\) are differentiable, \(u(1, 2) = 5\), \(u_s(1, 2) = 4\), \(u_t(1, 2) = -3\), \(v(1, 2) = 7\), \(v_s(1, 2) = 2\), \(v_t(1, 2) = 6\), \(G_u(5, 7) = 9\), \(G_v(5, 7) = -2\). Find \(R_s(1, 2)\) and \(R_t(1, 2)\).

Suppose \(f\) is a differentiable function of \(x\) and \(y\), and \(g(u, v) = f(e^u + \sin v, e^u + \cos v)\). Use the table of values to calculate \(g_u(0, 0)\) and \(g_v(0, 0)\).

Suppose \(f\) is a differentiable function of \(x\) and \(y\), and \(g(r, s) = f(2r - s, s^2 - 4r)\). Use the table of values in Exercise 15 to calculate \(g_r(1, 2)\) and \(g_s(1, 2)\).

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. \(u = f(x, y)\), where \(x = x(r, s, t)\), \(y = y(r, s, t)\)

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. \(w = f(x, y, z)\), where \(x = x(u, v)\), \(y = y(u, v)\), \(z = z(u, v)\)

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. \(T = F(p, q, r)\), where \(p = p(x, y, z)\), \(q = q(x, y, z)\), \(r = r(x, y, z)\)

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. \(R = F(t, u)\) where \(t = t(w, x, y, z)\), \(u = u(w, x, y, z)\)

Use the Chain Rule to find the indicated partial derivatives. \(z = x^4 + x^2y\), \(x = s + 2t - u\), \(y = stu^2\); find \(\frac{\partial z}{\partial s}\), \(\frac{\partial z}{\partial t}\), \(\frac{\partial z}{\partial u}\) when \(s = 4, t = 2, u = 1\).

Use the Chain Rule to find the indicated partial derivatives. \(T = \frac{v}{2u+v}\), \(u = p q \sqrt{r}\), \(v = p \sqrt{q} r\); find \(\frac{\partial T}{\partial p}\), \(\frac{\partial T}{\partial q}\), \(\frac{\partial T}{\partial r}\) when \(p = 2, q = 1, r = 4\).

Use the Chain Rule to find the indicated partial derivatives. \(w = xy + yz + zx\), \(x = r \cos \theta\), \(y = r \sin \theta\), \(z = r\theta\); find \(\frac{\partial w}{\partial r}\), \(\frac{\partial w}{\partial \theta}\) when \(r = 2, \theta = \pi/2\).

Use the Chain Rule to find the indicated partial derivatives. \(P = \sqrt{u^2 + v^2 + w^2}\), \(u = xe^y\), \(v = ye^x\), \(w = e^{xy}\); find \(\frac{\partial P}{\partial x}\), \(\frac{\partial P}{\partial y}\) when \(x = 0, y = 2\).

Use the Chain Rule to find the indicated partial derivatives. \(N = \frac{p+q}{p+r}\), \(p = u + vw\), \(q = v + uw\), \(r = w + uv\); find \(\frac{\partial N}{\partial u}\), \(\frac{\partial N}{\partial v}\), \(\frac{\partial N}{\partial w}\) when \(u = 2, v = 3, w = 4\).

Use the Chain Rule to find the indicated partial derivatives. \(u = x^2e^{y\tau}\), \(x = \alpha^2\beta\), \(y = \beta^2\gamma\), \(\tau = \gamma^2\alpha\); find \(\frac{\partial u}{\partial \alpha}\), \(\frac{\partial u}{\partial \beta}\), \(\frac{\partial u}{\partial \gamma}\) when \(\alpha = -1, \beta = 2, \gamma = 1\).

Use Equation 6 to find \(dy/dx\). \(y \cos x = x^2 + y^2\)

Find \(dy/dx\). \(\cos(xy) = 1 + \sin y\)

Find \(dy/dx\). \(\tan^{-1}(x^2y) = x + xy^2\)

Find \(dy/dx\). \(e^y \sin x = x + xy\)

Find \(\partial z/\partial x\) and \(\partial z/\partial y\). \(x^2 + 2y^2 + 3z^2 = 1\)

Find \(\partial z/\partial x\) and \(\partial z/\partial y\). \(x^2 - y^2 + z^2 - 2z = 4\)

Find \(\partial z/\partial x\) and \(\partial z/\partial y\). \(e^z = xyz\)

Find \(\partial z/\partial x\) and \(\partial z/\partial y\). \(yz + x \ln y = z^2\)

The temperature at a point \((x, y)\) is \(T(x, y)\), measured in degrees Celsius. A bug crawls so that its position after \(t\) seconds is given by \(x = \sqrt{1 + t}\), \(y = 2 + \frac{1}{3}t\), where \(x\) and \(y\) are measured in centimeters. The temperature function satisfies \(T_x(2, 3) = 4\) and \(T_y(2, 3) = 3\). How fast is the temperature rising on the bug’s path after 3 seconds?

Wheat production \(W\) in a given year depends on the average temperature \(T\) and the annual rainfall \(R\). Scientists estimate that the average temperature is rising at a rate of 0.15°C/year and rainfall is decreasing at a rate of 0.1 cm/year. They also estimate that at current production levels, \(\partial W / \partial T = -2\) and \(\partial W / \partial R = 8\). (a) What is the significance of the signs of these partial derivatives? (b) Estimate the current rate of change of wheat production, \(dW/dt\).

The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation \(C = 1449.2 + 4.6T - 0.055T^2 + 0.00029T^3 + 0.016D\) where \(C\) is the speed of sound (in meters per second), \(T\) is the temperature (in degrees Celsius), and \(D\) is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver’s depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units?

The length \(l\), width \(w\), and height \(h\) of a box change with time. At a certain instant the dimensions are \(l = 1\) m and \(w = h = 2\) m, and \(l\) and \(w\) are increasing at a rate of 2 m/s while \(h\) is decreasing at a rate of 3 m/s. At that instant find the rates at which the following quantities are changing. (a) The volume (b) The surface area (c) The length of a diagonal

The voltage \(V\) in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance \(R\) is slowly increasing as the resistor heats up. Use Ohm’s Law, \(V = IR\), to find how the current \(I\) is changing at the moment when \(R = 400\ \Omega\), \(I = 0.08\) A, \(dV/dt = -0.01\) V/s, and \(dR/dt = 0.03\ \Omega\)/s.

The pressure of 1 mole of an ideal gas is increasing at a rate of 0.05 kPa/s and the temperature is increasing at a rate of 0.15 K/s. Use the equation \(PV = 8.31T\) in Example 2 to find the rate of change of the volume when the pressure is 20 kPa and the temperature is 320 K.

A manufacturer has modeled its yearly production function \(P\) (the value of its entire production, in millions of dollars) as a Cobb-Douglas function \(P(L, K) = 1.47L^{0.65}K^{0.35}\) where \(L\) is the number of labor hours (in thousands) and \(K\) is the invested capital (in millions of dollars). Suppose that when \(L = 30\) and \(K = 8\), the labor force is decreasing at a rate of 2000 labor hours per year and capital is increasing at a rate of $500,000 per year. Find the rate of change of production.

One side of a triangle is increasing at a rate of 3 cm/s and a second side is decreasing at a rate of 2 cm/s. If the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is 20 cm long, the second side is 30 cm, and the angle is \(\pi/6\)?

A sound with frequency \(f_s\) is produced by a source traveling along a line with speed \(v_s\). If an observer is traveling with speed \(v_o\) along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is \(f_o = \left(\frac{c + v_o}{c - v_s}\right) f_s\) where \(c\) is the speed of sound, about 332 m/s. (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 34 m/s and accelerating at 1.2 m/s\(^2\). A train is approaching you from the opposite direction on the other track at 40 m/s, accelerating at 1.4 m/s\(^2\), and sounds its whistle, which has a frequency of 460 Hz. At that instant, what is the perceived frequency that you hear and how fast is it changing?

Assume that all the given functions are differentiable. If \(z = f(x, y)\), where \(x = r \cos \theta\) and \(y = r \sin \theta\), (a) find \(\partial z/\partial r\) and \(\partial z/\partial \theta\) and (b) show that \[ \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = \left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2 \]

Assume that all the given functions are differentiable. If \(u = f(x, y)\), where \(x = e^s \cos t\) and \(y = e^s \sin t\), show that \[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 = e^{-2s}\left[\left(\frac{\partial u}{\partial s}\right)^2 + \left(\frac{\partial u}{\partial t}\right)^2\right] \]

Assume that all the given functions are differentiable. If \(z = \frac{1}{x}[f(x - y) + g(x + y)]\), show that \[ \frac{\partial}{\partial x}\left(x^2 \frac{\partial z}{\partial x}\right) = x^2 \frac{\partial^2 z}{\partial y^2} \]

Assume that all the given functions are differentiable. If \(z = \frac{1}{y}[f(ax + y) + g(ax - y)]\), show that \[ \frac{\partial}{\partial x}\left(y^2 \frac{\partial z}{\partial x}\right) = a^2 \frac{\partial}{\partial y}\left(y^2 \frac{\partial z}{\partial y}\right) \]

Assume that all the given functions have continuous second-order partial derivatives. Show that any function of the form \(z = f(x + at) + g(x - at)\) is a solution of the wave equation \[ \frac{\partial^2 z}{\partial t^2} = a^2 \frac{\partial^2 z}{\partial x^2} \] [Hint: Let \(u = x + at, v = x - at\).]

Assume that all the given functions have continuous second-order partial derivatives. If \(u = f(x, y)\), where \(x = e^s \cos t\) and \(y = e^s \sin t\), show that \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = e^{-2s}\left[\frac{\partial^2 u}{\partial s^2} + \frac{\partial^2 u}{\partial t^2}\right] \]

Assume that all the given functions have continuous second-order partial derivatives. If \(z = f(x, y)\), where \(x = r^2 + s^2\) and \(y = 2rs\), find \(\partial^2 z/\partial r \partial s\). (Compare with Example 7.)

Assume that all the given functions have continuous second-order partial derivatives. If \(z = f(x, y)\), where \(x = r \cos \theta\) and \(y = r \sin \theta\), find (a) \(\partial z/\partial r\), (b) \(\partial z/\partial \theta\), and (c) \(\partial^2 z/\partial r \partial \theta\).

Assume that all the given functions have continuous second-order partial derivatives. If \(z = f(x, y)\), where \(x = r \cos \theta\) and \(y = r \sin \theta\), show that \[ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = \frac{\partial^2 z}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 z}{\partial \theta^2} + \frac{1}{r}\frac{\partial z}{\partial r} \]

Assume that all the given functions have continuous second-order partial derivatives. Suppose \(z = f(x, y)\), where \(x = g(s, t)\) and \(y = h(s, t)\). (a) Show that \[ \frac{\partial^2 z}{\partial t^2} = \frac{\partial^2 z}{\partial x^2}\left(\frac{\partial x}{\partial t}\right)^2 + 2\frac{\partial^2 z}{\partial x \partial y}\frac{\partial x}{\partial t}\frac{\partial y}{\partial t} + \frac{\partial^2 z}{\partial y^2}\left(\frac{\partial y}{\partial t}\right)^2 + \frac{\partial z}{\partial x}\frac{\partial^2 x}{\partial t^2} + \frac{\partial z}{\partial y}\frac{\partial^2 y}{\partial t^2} \] (b) Find a similar formula for \(\partial^2 z/\partial s \partial t\).

A function \(f\) is called homogeneous of degree n if it satisfies the equation \(f(tx, ty) = t^n f(x, y)\) for all \(t\), where \(n\) is a positive integer and \(f\) has continuous second-order partial derivatives. (a) Verify that \(f(x, y) = x^2y + 2xy^2 + 5y^3\) is homogeneous of degree 3. (b) Show that if \(f\) is homogeneous of degree \(n\), then \[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = nf(x, y) \] [Hint: Use the Chain Rule to differentiate \(f(tx, ty)\) with respect to \(t\).]

If \(f\) is homogeneous of degree \(n\), show that \[ x^2 \frac{\partial^2 f}{\partial x^2} + 2xy \frac{\partial^2 f}{\partial x \partial y} + y^2 \frac{\partial^2 f}{\partial y^2} = n(n-1)f(x, y) \]

If \(f\) is homogeneous of degree \(n\), show that \(f_x(tx, ty) = t^{n-1}f_x(x, y)\)

Suppose that the equation \(F(x, y, z) = 0\) implicitly defines each of the three variables \(x, y\), and \(z\) as functions of the other two: \(z = f(x, y)\), \(y = g(x, z)\), \(x = h(y, z)\). If \(F\) is differentiable and \(F_x, F_y\), and \(F_z\) are all nonzero, show that \[ \frac{\partial z}{\partial x} \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} = -1 \]

Equation 6 is a formula for the derivative \(dy/dx\) of a function defined implicitly by an equation \(F(x, y) = 0\), provided that \(F\) is differentiable and \(F_y \ne 0\). Prove that if \(F\) has continuous second derivatives, then a formula for the second derivative of \(y\) is \[ \frac{d^2 y}{dx^2} = - \frac{F_{xx}F_y^2 - 2F_{xy}F_x F_y + F_{yy}F_x^2}{F_y^3} \]