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Juan Hewlett

Answered question

2021-12-09

Find the first and second derivatives of the function

f (x) = {(x^{2} + 4)}^{5}

Answer & Explanation

Neil Dismukes

Beginner2021-12-10Added 37 answers

So,
$f^{'} (x) = \frac{d}{d x} ({(x^{2} + 4)}^{5})$
$= 5 {(x^{2} + 4)}^{5 - 1} \cdot \frac{d}{d x} (x^{2} + 4)$ (By chain rule and $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ )
$= 5 {(x^{2} + 4)}^{4} \cdot (2 x)$
$= 10 x {(x^{2} + 4)}^{4}$
$f (x) = {(x^{2} + 4)}^{5} i s f^{'} (x) = 10 x {(x^{2} + 4)}^{4}$ .
$f^{″} (x) = \frac{d}{d x} (10 x {(x^{2} + 4)}^{4})$
$= 10 \frac{d}{d x} (x {(x^{2} + 4)}^{4})$
$= 10 ({(x^{2} + 4)}^{4} \cdot \frac{d}{d x} (x) + x \cdot \frac{d}{d x} ({(x^{2} + 4)}^{4}))$
$= 10 ({(x^{2} + 4)}^{4} \cdot (1) + x \cdot (4 {(x^{2} + 4)}^{4 - 1} \cdot \frac{d}{d x} (x^{2} + 4)))$
(Using the chain rule, $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ )
$= 10 ({(x^{2} + 4)}^{4} + x \cdot (4 {(x^{2} + 4)}^{3} \cdot (2 x)))$

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Find the first and second derivatives of the function f(x)=(x^{2}+4)^{5}

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