AP Calculus BC Fun 3 C Calculate Derivatives of Compositions of Differentiable Functions Questions

Practice AP Calculus BC questions on applying the chain rule to calculate derivatives of compositions of differentiable functions.

[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}

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AP Calculus BC Fun 3 C Calculate Derivatives of Compositions of Differentiable Functions Questions question 1