AP Calculus BC Unit 3: Derivative Rules

Practice AP Calculus BC Unit 3 questions on applying the chain rule and derivative rules to composite and higher-order derivatives.

Covered syllabus points

[Maximum number: 3]

Consider the family of functions $f(x)=\frac{1}{x^{2}-2 x+k}$ , where k is a constant.

Find the value of k, for k>0, such that the slope of the line tangent to the graph of f at x=0 equals 6.

\begin{aligned} & f^{\prime}(x)=\frac{-(2 x-2)}{\left(x^{2}-2 x+k\right)^{2}} \\ & f^{\prime}(0)=\frac{2}{k^{2}}=6 \Rightarrow k^{2}=\frac{1}{3} \Rightarrow k=\frac{1}{\sqrt{3}} \end{aligned}

[Maximum number: 1]

If y is a differentiable function of x, then the slope of the curve of $x y^{2}$ $-2 y+4 y^{3}=6$ at the point where y=1 is

$-\frac{1}{18}$

$\frac{5}{18}$

$\frac{1}{3}$

[Maximum number: 1]

The table above gives values of differentiable functions f and g. If $H(x)=f^{-1}(x)$ , then $H^{\prime}(3)$ equals

$-\frac{1}{16}$

$-\frac{1}{8}$

$\frac{1}{2}$

[Maximum number: 1]

If $f(x)=\sin x+2 x+1$ and g is the inverse function of f, what is the value of $g^{\prime}(1)$ ?

$\frac{1}{3}$

$\frac{1}{2+\cos 1}$

$2+\cos 1$

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