Q: Let $$y = \sec(x) \tan(x). $$What is $$\frac{dy}{dx}$$?

$$\sec(x) \tan^2$(x) + \sec^3$(x)$$

Question 1

Suppose $$h(x) = f(g(x))$$ where $$f$$ and $$g$$ are differentiable functions. Which of the following correctly expresses $$h'(x)$$ using the Chain Rule?

Accepted Answer

$$h'(x) = f'(g(x)) \cdot g'(x)$$

Question 2

Let $$y = \sin(e$$^{3x}$$). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$3e^{3x}$$ \$$cos(e^{3x}$$)$$

Question 3

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{-0.5t^2}$$, where t$ is in hours. What is the rate of change of concentration with respect to time at t=2$?

Accepted Answer

Question 4

Let $$y = \cos(\sin(x)). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\sin(\sin(x)) \cdot \cos(x)$$

Question 5

Suppose $$f(x)$$ and $$g(x)$$ are differentiable, and $$h(x) = g(f(x)). If$$ $$f(1) = 2$$, $$g'(2) = 5$$, and $$f'(1) = 3$$, what is $$h'(1)$$?

Accepted Answer

$$h'(1) = 5 \cdot 3 = 15$$

Question 6

Let $$y = \sec(x^2$). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x \sec(x^2$) 	an(x^2$)$$

Question 7

A population grows according to $$P(t) = 1000e$$^{0.02t^2}$$. What is the instantaneous rate of growth at t=5$?

Accepted Answer

$$P'(5) = 1000e$$^{0.5}$$ \cdot 0.04 \cdot 5$$

Question 8

Let $$y = 	an(3x + 5). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$3 \sec^2$(3x + 5)$$

Question 9

Suppose $$y = e$$^{\sin(x)}$$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{\sin(x)}$$ \cos(x)$$

Question 10

Let $$y = \csc(\cos(x)). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\csc(\cos(x)) \cot(\cos(x)) \sin(x)$$

Question 11

A chemical reaction's rate is given by $$R(t) = \sin(e$$^{t}$$). $$What is $$R'(t)$$?

Accepted Answer

$$e^{t}$$ \$$cos(e^{t}$$)$$

Question 12

Let $$y = \sec(x) 	an(x). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec(x) 	an^2$(x) + \sec^3$(x)$$

Question 13

Suppose $$y = \cos(e$$^{x^2}$$). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$-2x e$$^{x^2}$$ \sin(e$$^{x^2}$$)$$

Question 14

Let $$y = (x^2$ + 1)^5$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$10x(x^2$$ + $$1)^4$$

Question 15

Suppose $$y = \sin^2$(x^3$$)$$. What is $$\frac{dy}{dx}$?

Accepted Answer

$$6x^2$$ \$$sin(x^3$$) \$$cos(x^3$$)$$

Question 16

A temperature in a chemical process is modeled by $$T(t) = \cos^2$(e^{t}$$)$$. What is $$\frac{dT}{dt}$?

Accepted Answer

$$-2e^{t}$$ \$$cos(e^{t}$$) \$$sin(e^{t}$$)$$

Question 17

Let $$y = \ln(\sin(x^2$)). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2x \cos(x^2$)}{\sin(x^2$)}$$

Question 18

Suppose $$y = e$$^{e^{x}$$}. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{e^{x}$$} \cdot $$e^{x}$$

Question 19

Let $$y = \sin(\cos(3x)). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

The Chain Rule: Advanced Applications and Examples

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