Q: Let $$h(x) = f(g(x))$$ where $$f$$ and $$g$$ are differentiable functions. If $$f'(2) = 5$$, $$g(3) = 2$$, and $$g'(3) = -4$$, what is $$h'(3)$$?

$$h'(3) = 5 \times (-4) = -20$$

Q: Let $$y = \sec(x^2$) \cdot (x + 9). $$Find $$\frac{dy}{dx}.$$

$$\frac{dy}{dx} = \sec(x^2$) \tan(x^2$) \cdot 2x \cdot (x+9) + \sec(x^2$)$$

Question 1

Let $$h(x) = f(g(x))$$ where $$f$$ and $$g$$ are differentiable functions. If $$f'(2) = 5$$, $$g(3) = 2$$, and $$g'(3) = -4$$, what is $$h'(3)$$?

Accepted Answer

$$h'(3) = 5 	imes (-4) = -20$$

Question 2

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

Accepted Answer

$$\frac{dy}{dx} = \cos(e$$^{3x}$$) \cdot 3e$$^{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 at t = 2$ hours?

Accepted Answer

$$C'(2) = -2 \cdot 2 \cdot 0.5 \cdot e$$^{-0.5 \cdot 4}$$ = -2e$$^{-2}$$

Question 4

Let $$y = \sec(x^2$) \cdot (x + 9). $$Find $$\frac{dy}{dx}.$$

Accepted Answer

$$\frac{dy}{dx} = \sec(x^2$) 	an(x^2$) \cdot 2x \cdot (x+9) + \sec(x^2$)$$

Question 5

Given $$y = \cos(\sin(x+5))$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = -\sin(\sin(x+5)) \cdot \cos(x+5)$$

Question 6

If $$f(x) = e$$^{e^x}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = e$$^{e^x}$$ \cdot e^x$$

Question 7

A population of bacteria grows according to $$P(t) = 3$$^{2t+1}$$. What is $$\frac{dP}{dt}$$ at t=0$?

Accepted Answer

$$\frac{dP}{dt} = 3$$^{2t+1}$$ \cdot 2 \ln 3$$;\ at\ t=0:\ $$3^1$$ \cdot 2 \ln 3 = 6 \ln 3$

Question 8

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

Accepted Answer

$$\frac{dy}{dx} = \sec^2$(3x^5$$) \cdot $$15x^4$

Question 9

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

Accepted Answer

$$\frac{dy}{dx} = e$$^{\sin x}$$ \cdot \cos x$$

Question 10

A chemical reaction's rate is given by $$R(t) = \cos(e$$^{t^2}$$). $$What is $$R'(t)$$?

Accepted Answer

$$R'(t) = -\sin(e$$^{t^2}$$) \cdot e$$^{t^2}$$ \cdot 2t$$

Question 11

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

Accepted Answer

$$\frac{dy}{dx} = -\csc(\cos(x+5)) \cot(\cos(x+5)) \cdot (-\sin(x+5))$$

Question 12

If $$y = \sec(x^2$) 	an(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \sec(x^2$) 	an(x^2$) \cdot 2x [\sec^2$(x^2$$) + \$$tan(x^2$$)]$

Question 13

Suppose $$f(x) = \sin(x^3$). $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = \cos(x^3$) \cdot 3x^2$

Question 14

A function is defined as $$y = e$$^{x^2 + 3x}$$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = e$$^{x^2 + 3x}$$ (2x + 3)$$

Question 15

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

Accepted Answer

$$\frac{dy}{dx} = 2\sin(e^x)\cos(e^x) \cdot e^x$$

Question 16

If $$y = 	an(e$$^{3x+1}$$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \sec^2$(e^{3x+1}$$) \cdot $$e^{3x+1}$$ \cdot 3$

Question 17

A radioactive substance decays according to $$A(t) = e$$^{-kt^2}$$, where k$$ \u003e 0$$. What is A$$'(t)$$?

Accepted Answer

$$A'(t) = -2kt e$$^{-kt^2}$$

Question 18

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

Accepted Answer

$$\frac{dy}{dx} = 2\cos(x^2$)(-\sin(x^2$)) \cdot 2x = -4x\cos(x^2$)\sin(x^2$)$$

Question 19

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

Accepted Answer

$$\frac{dy}{dx} = \cos(\cos(2x)) \cdot (-\sin(2x)) \cdot 2$$

The Chain Rule: Advanced Applications and Examples

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