Question 1

The diagram above illustrates the process of function composition. If $$f(x)$$ and $$g(x)$$ are differentiable functions, which rule should be used to find the derivative of $$f(g(x))$$ with respect to $$x$?

Accepted Answer

The Chain Rule

Question 2

Given the composition of functions as shown in the diagram, if $$f(x) = x^2$ and g(x) = 3x + 1$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$(3x + 1)^2$

Question 3

If $$y = f(g(x))$$ and $$f'(u) = 2u$$, $$g(x) = x^3$, what is $$\frac{dy}{dx}$$ at x = 2$?

Accepted Answer

$$48$$

Question 4

Suppose $$f(x)$$ models the temperature in a chemical reaction as a function of concentration $$x$$, and $$g(x)$$ models the concentration as a function of time $$x. $$What does the composition $$f(g(x))$$ represent?

Accepted Answer

The temperature as a function of time

Question 5

If $$f(x) = \sin(x)$$ and $$g(x) = x^2$, what is $$\frac{d}{dx} f(g(x))$$?

Accepted Answer

$$2x \cos(x^2$)$$

Question 6

Given $$f(x) = e^x$$ and $$g(x) = 4x - 5$$, find $$\frac{d}{dx} f(g(x)).$$

Accepted Answer

$$4e^{4x-5}$$

Question 7

If $$h(x) = f(g(x))$$ and $$f'(x) = 3x^2$, g(x) = x + 2$, what is h$$'(x)$$?

Accepted Answer

$$3(x+2)^2$$

Question 8

A company’s profit $$P$$ depends on the number of units sold $$q$$, which in turn depends on the amount spent on advertising $$A. If$$ $$P(q)$$ and $$q(A)$$ are differentiable, what does $$\frac{dP}{dA}$$ represent?

Accepted Answer

The rate at which profit changes with respect to advertising

Question 9

If $$f(x) = \ln(x)$$ and $$g(x) = x^2$ + 1$$, what is $$\frac{d}{dx} f(g(x))$$?

Accepted Answer

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

Question 10

Suppose $$f(x) = (3x + 2)^5$. Using the chain rule, what is f$$'(x)$$?

Accepted Answer

$$5(3x + 2)^4$ \cdot 3$$

Business Calculus Study Notes: Techniques of Differentiation and Applications

Study Guide - Practice Questions

Study Guide - Flashcards