Composite, implicit and inverse functions practice questions (AP Calculus)

Sample questions

All 2 free AP Calculus questions on this unit, from the free AP Calculus AB Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Chain rule
   If $y = \sin(x^2)$, then $\dfrac{dy}{dx} =$
   • A.
     $\cos(x^2)$
   • B.
     $2x\cos(x^2)$
   • C.
     $2x\cos(2x)$
   • D.
     $-2x\cos(x^2)$
   Show answerFrom the free AP Calculus AB Practice Exam →
2. Question 22 marks · Implicit differentiation
   The curve $x^2 + y^2 = 25$ passes through the point $(3, 4)$. The slope of the curve at that point is:
   • A.
     $\frac{3}{4}$
   • B.
     $-\frac{3}{4}$
   • C.
     $\frac{4}{3}$
   • D.
     $-\frac{4}{3}$
   Show answerFrom the free AP Calculus AB Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for composite, implicit and inverse functions, typeset as they appear on the paper.

Chain rule

Differentiate the outer function at the inner function, then multiply by the derivative of the inner function. Forgetting the inner derivative is the most common error on the exam.

$$\frac{d}{dx} f\big(g(x)\big) = f'\big(g(x)\big)\, g'(x)$$

Implicit differentiation

For a curve given by an equation in $x$ and $y$, differentiate both sides with respect to $x$, attaching $\frac{dy}{dx}$ to every term differentiated in $y$, then solve for $\frac{dy}{dx}$.

Derivative of an inverse function

The slope of $f^{-1}$ at a point is the reciprocal of the slope of $f$ at the corresponding point. Table questions test this: find the $x$ with $f(x) = b$, then take $\frac{1}{f'(x)}$.

$$\left(f^{-1}\right)'(b) = \frac{1}{f'\big(f^{-1}(b)\big)}$$

Inverse trigonometric derivatives

The three the exam expects from memory; arccos and arccot are the negatives of arcsin and arctan.

$$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}, \qquad \frac{d}{dx}\arctan x = \frac{1}{1 + x^2}, \qquad \frac{d}{dx}\text{arcsec}\, x = \frac{1}{|x|\sqrt{x^2 - 1}}$$

Higher-order derivatives

The second derivative is the derivative of the first; in motion problems it is acceleration, in curve analysis it gives concavity. For implicit curves, $\frac{d^2y}{dx^2}$ requires substituting the expression for $\frac{dy}{dx}$.

$$f''(x) = \frac{d}{dx}\big(f'(x)\big) = \frac{d^2 y}{dx^2}$$

Derivatives of exponentials and logarithms with other bases

Rewrite with $e$ and $\ln$: $a^x = e^{x \ln a}$ and $\log_a x = \frac{\ln x}{\ln a}$.

$$\frac{d}{dx} a^x = a^x \ln a, \qquad \frac{d}{dx} \log_a x = \frac{1}{x \ln a}$$

Frequently asked questions

Other AP Calculus topics

• Limits and continuity
• Differentiation: definition and basic rules
• Contextual applications of differentiation
• Analytical applications of differentiation
• Integration and accumulation of change
• Differential equations
• Applications of integration
• Parametric, polar and vector functions (BC)
• Infinite sequences and series (BC)

All AP Calculus topics · AP Calculus exam guide · All free practice exams