Q: A company invests $$50,000 at 1.25% annual interest, compounded continuously. The account value after t$ years is A(t) = 50$$,$$000e^{0.0125t}. $$What is the rate of growth of the account at $$t = 5$ years?

$$A'(5) = 625e$$^{0.0125 \times 5}$$

Question 1

Which of the following is the derivative of the function $$f(x) = e$$^{x}$$?

Accepted Answer

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

Question 2

Given $$y = 3e$$^{x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$3e^{x}$$

Question 3

If $$y = x$$^{2}$$e^{x}$$, what is $$\frac{dy}{dx}$?

Accepted Answer

$$e^{x}(x$$^{2}$$ + 2x)$

Question 4

For any differentiable function $$u(x)$$, what is $$\frac{d}{dx} e$$^{u(x)}$$?

Accepted Answer

$$e^{u(x)}$$ \cdot u'(x)$$

Question 5

Differentiate $$y = e$$^{8x}$$ with respect to x$.

Accepted Answer

$$8e^{8x}$$

Question 6

Find $$\frac{d}{dx} \left( e$$^{-4x}$$ \right).$$

Accepted Answer

$$-4e^{-4x}$$

Question 7

If $$y = e$$^{x^{2}$$ + 4x - 7}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{x^{2} + 4x - 7$$} \cdot (2x + 4)$$

Question 8

A company invests $$50,000 at 1.25% annual interest, compounded continuously. The account value after t$ years is A(t) = 50$$,$$000e^{0.0125t}. $$What is the rate of growth of the account at $$t = 5$ years?

Accepted Answer

$$A'(5) = 625e$$^{0.0125 	imes 5}$$

Question 9

Find $$\frac{d}{dx} \left( e$$^{\sqrt{x^{2}$$ - 3}} \right).$$

Accepted Answer

$$\frac{x e$$^{\sqrt{x^{2}$$ - 3}}}{\sqrt{x$$^{2}$$ - 3}}$$

Question 10

If $$y = e$$^{-5x^{2}$$}$$, what is $$\frac{d$$^{2}$$y}{dx$$^{2}$$}$$?

Accepted Answer

$$10e^{-5x^{2}$$}$$(10x^{2} - 1$$)$$

Question 11

Which rule is used to differentiate $$e^{f(x)}$$ where $$f(x) is a $$differentiable function?

Accepted Answer

Chain Rule

Question 12

If $$y = x$$^{3}$$e^{x}$$, what is $$\frac{dy}{dx}$?

Accepted Answer

$$3x^{2}e$$^{x}$$ + x$$^{3}$$e^{x}$$

Question 13

Differentiate $$y = e$$^{x^{3}$$ + 8x}$$ with respect to $$x.$$

Accepted Answer

$$e^{x^{3} + 8x$$}$$(3x^{2} + 8$$)$$

Question 14

If $$y = \frac{e$$^{x}$$}{x$$^{3}$$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{x$$^{3}$$e^{x}$$ - $$e^{x}$$ \cdot $$3x^{2}$$}{$$x^{6}$$}$

Question 15

A business account grows according to $$A(t) = 50,000e$$^{0.0125t}$$. What does A$$'(t)$$ represent in this context?

Accepted Answer

The rate at which the account value is changing at time $$t$$

Question 16

Find $$\frac{d}{dx} \left( e$$^{x^{2}$$ + 8x} \right).$$

Accepted Answer

$$e^{x^{2} + 8x$$}(2x + 8)$$

Question 17

If $$y = e$$^{\sqrt{x^{2}$$ + 5}}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{x e$$^{\sqrt{x^{2}$$ + 5}}}{\sqrt{x$$^{2}$$ + 5}}$$

Question 18

Which of the following best describes the derivative of $$e^{x}$$?

Accepted Answer

It is always equal to $$e^{x}$$ for any $$x$$

Derivatives of Exponential (Base-e) Functions and Business Applications

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