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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