Question 1

Let $$f(x) = a^x$$ where $$a \u003e 0. $$What is the derivative $$f'(x)$$?

Accepted Answer

$$(\ln a) \cdot a^x$$

Question 2

Suppose a population grows according to the logistic function $$P(t) = \frac{mP_0$}{P_0$ + (m - P_0$)e^{-kmt}$$}$$. What is the limiting value of P(t$$)$$ as t$$ 	o \infty$?

Accepted Answer

$$m$$

Question 3

Find $$\frac{d}{dx}(e$$^{x^2}$$).$$

Accepted Answer

$$2x e$$^{x^2}$$

Question 4

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 5

Let $$f(x) = 3x^2$ \cdot \log_3$ x. $$What is $$f'(x)$$?

Accepted Answer

$$6x \cdot \log_3$ x + \frac{3x^2$}{x \ln 3}$$

Question 6

If $$y = \ln(kx)$$ where $$k 
eq 0 is a $$constant, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x}$$

Question 7

Find the derivative of $$f(x) = a$$^{g(x)}$$ where a$$ \u003e 0$$ and g(x$$)$$ is differentiable.

Accepted Answer

$$(\ln a) \cdot a$$^{g(x)}$$ \cdot g'(x)$$

Question 8

What is the derivative of $$f(x) = \log_a x$$ for $$x \u003e 0$$ and $$a \u003e 0$$, $$a 
eq 1$$?

Accepted Answer

$$\frac{1}{x \ln a}$$

Question 9

Suppose $$f(x) = \ln|x|. $$What is $$f'(x)$$ for $$x 
eq 0$$?

Accepted Answer

$$\frac{1}{x}$$

Question 10