Question 1

Suppose $$x = f(y)$$ and $$y = g(t)$$, where both $$f$$ and $$g$$ are differentiable functions. According to the chain rule, what is $$\frac{dx}{dt}$$?

Accepted Answer

$$\frac{dx}{dt} = \frac{dx}{dy} \cdot \frac{dy}{dt}$$

Question 2

Let $$x = y^2$ + 2xy^4$, where x$$ = \sin(2t)$$ and y$$ = \cos t$$. Find $$\frac{dx}{dt}$$ at t = 0$.

Accepted Answer

$$-2$$

Question 3

Given $$x = \sqrt{t^2$ - 1}$$, $$y = e^x$$, and $$z = e^y$$, find $$\frac{dz}{dt}$$ using the chain rule.

Accepted Answer

$$\frac{dz}{dt} = \frac{t}{\sqrt{t^2$ - 1}} e$$^{e^{\sqrt{t^2 - 1}$$}}$$

Question 4

If $$z = e$$^{y \sin x}$$, where x$$ = $$s^2$$ and $$y = s^3$, find $$\frac{dz}{ds}$$.

Accepted Answer

$$e^{s^3 \sin s^2}$$ \left( $$3s^2$$ \sin $$s^2$$ + $$2s^3$$ \cos $$s^2$$ \right)$$

Question 5

Consider $$x = 3x^2$ - 2xy$$, $$u = e$$^{x}$$ \sin y$$, $$y = e$$^{t}$$ \cos t. $$Find $$\frac{du}{dt}$$ using the chain rule.

Accepted Answer

$$e^{x}$$ \sin y \frac{dx}{dt} + $$e^{x}$$ \cos y \frac{dy}{dt}$$

Question 6

In the context of implicit differentiation, if $$F(x, y) = 0$$ defines $$y$$ implicitly as a function of $$x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = -\frac{F_x}{F_y}$$

Question 7

Suppose $$z is $$given implicitly as a function $$z = f(x, y) by $$the equation $$F(x, y, z) = 0. $$What is $$\frac{\partial z}{\partial x}$$ using the chain rule?

Accepted Answer

$$\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$$

Question 8

Given $$x^4$$ + $$y^4$$ + $$z^4$$ + $$e^{2z} + 6$$xyz = 1$$, find $$\frac{\partial z}{\partial x}$$ at a point where x = 1$, y = 1$, z = 0$.

Accepted Answer

$$-\frac{4x^3$ + 6yz}{4z^3$ + 2e$$^{2z}$$ + 6xy}$$

Question 9

Which of the following is a correct application of the chain rule for $$\frac{dz}{dt} if$$ $$z = f(x, y)$$, $$x = g(t)$$, $$y = h(t)$$?

Accepted Answer

$$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$$

Question 10

A patient’s body temperature $$T (in$$ $$^\circC) $$changes with time $$t (in $$hours) according to $$T = 37 + 2e$$^{-0.5t}$$. If the rate of change of t$ with respect to time is 1$ hour per hour, what is $$\frac{dT}{dt}$$ at t = 2$?

Accepted Answer

$$-e^{-1}$$

Question 11

If $$y = x^2$ and x$$ = \sin t$$, what is $$\frac{dy}{dt}$$?

Accepted Answer

$$2 \sin t \cos t$$

Question 12

Let $$z = x^2$ y^3$, x$$ = $$t^2$$, $$y = e^t. $$Find $$\frac{dz}{dt}.$$

Accepted Answer

$$2t e$$^{3t}$$ t^4$ + 3e$$^{2t}$$ t^2$

Question 13

A chemical reaction rate $$R$$ depends on temperature $$T$$ and concentration $$C as$$ $$R = kTC$$, where $$T = 300 + 10t$$ and $$C = 2e$$^{-t}$$. Find $$\frac{dR}{dt}$$ at t = 0$ if k = 1$.

Accepted Answer

Question 14

If $$z = f(x, y)$$, $$x = g(s, t)$$, $$y = h(s, t)$$, which formula gives $$\frac{\partial z}{\partial s}$$?

Accepted Answer

$$\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}$$

Question 15

Suppose $$F(x, y, z) = 0$$ and $$z is a $$function of $$x$$ and $$y. $$Which formula gives $$\frac{\partial z}{\partial y}$$?

Accepted Answer

$$\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$$

Question 16

A population $$P$$ grows according to $$P = e$$^{kt}$$, where k$ is a constant. If k = 0.03$, what is $$\frac{dP}{dt}$$ at t = 10$?

Accepted Answer

$$0.03 e$$^{0.3}$$

Question 17

Let $$z = x^2$ + y^2$, x$$ = $$t^3$$, $$y = \sin t. $$Find $$\frac{dz}{dt} at$$ $$t = \pi.$$

Accepted Answer

$$2t^3$$ \cdot $$3t^2 + 2$$\sin t \cdot \cos t$$ evaluated at t$$ = \pi$$

Section 4.5 - The Chain Rule and Implicit Differentiation

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