Fundamental Properties of Differentiation (College Board AP® Calculus AB)

Exam Questions

18 mins10 questions

3 marks

Function $h$ is a differentiable function with $h (5) = 2$ . The line $y = 5 + \frac{4}{5} (x - 7)$ is a tangent to the graph of $h$ at $x = 5$ .

Let $a$ be the function given by $a (x) = 4 x^{2} h (x)$ .

Write an expression for $a^{'} (x)$ and find $a^{'} (5)$ .

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2 marks

$x$	0	3	5	8
$f (x)$	-5	0	7	1
$f^{'} (x)$	-2	3	-4	8
$g (x)$	-6	2	-7	-1
$g^{'} (x)$	4	5	6	-8

The functions $f$ and $g$ are differentiable. The table shown gives the values of the functions and their first derivatives at selected values of $x$ .

Let $k$ be a differentiable function such that $k = {(f (x))}^{2} \times g (x)$ .

Find the value of $k' (5)$ .

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