Find the derivatives of h(x)=(−7(x2−8)2)2x

Question

SabadisO · Accepted Answer

Find the derivative of the following via implicit differentiation:

\frac{d}{d x} (h (x)) = \frac{d}{d x} (- 28 x (- 8 + x^{2}))

Using the chain rule,

\frac{d}{d x} (h (x)) = \frac{d h (u)}{d u} \frac{d u}{d x}

, where

u = x and \frac{d}{d u} (h (u)) = h^{'} (u)

:

(\frac{d}{d x} (x)) h^{'} (x) = \frac{d}{d x} (- 28 x (- 8 + x^{2}))

The derivative of x is 1:

1 h^{'} (x) = \frac{d}{d x} (- 28 x (- 8 + x^{2}))

Factor out constants:

h^{'} (x) = - 28 (\frac{d}{d x} (x (- 8 + x^{2})))

Use the product rule,

\frac{d}{d x} (u v) = v \frac{d u}{d x} + u \frac{d v}{d x}

, where

u = x and v = x^{2} - 8

:

h^{'} (x) = - 28 (- 8 + x^{2}) (\frac{d}{d x} (x)) + x (\frac{d}{d x} (- 8 + x^{2}))

The derivative of x is 1:

h^{'} (x) = - 28 (x (\frac{d}{d x} (- 8 + x^{2})) + 1 (- 8 + x^{2}))

Simplify the expression:

h^{'} (x) = - 28 (- 8 + x^{2} + x (\frac{d}{d x} (- 8 + x^{2})))

Differentiate the sum term by term:

h^{'} (x) = - 28 (- 8 + x^{2} + \frac{d}{d x} (- 8) + \frac{d}{d x} (x^{2}) x)

The derivative of -8 is zero:

Find the derivatives of {h}{\left({x}\right)}={\left(-{7}{\left({x}^{2}-{8}\right)}{2}\right)}{2}{x}