Find dy/dx y=(4x^2-14)^-12

$y = {(4 x^{2} - 14)}^{- 12}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(4 x^{2} - 14)}^{- 12})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 12}$ and $g (x) = 4 x^{2} - 14$ .

To apply the Chain Rule, set $u$ as $4 x^{2} - 14$ .

$\frac{d}{d u} [u^{- 12}] \frac{d}{d x} [4 x^{2} - 14]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 12$ .

$- 12 u^{- 13} \frac{d}{d x} [4 x^{2} - 14]$

Replace all occurrences of $u$ with $4 x^{2} - 14$ .

$- 12 {(4 x^{2} - 14)}^{- 13} \frac{d}{d x} [4 x^{2} - 14]$

Differentiate.

By the Sum Rule, the derivative of $4 x^{2} - 14$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 14]$ .

$- 12 {(4 x^{2} - 14)}^{- 13} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 14])$

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$- 12 {(4 x^{2} - 14)}^{- 13} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 12 {(4 x^{2} - 14)}^{- 13} (4 (2 x) + \frac{d}{d x} [- 14])$

Multiply $2$ by $4$ .

$- 12 {(4 x^{2} - 14)}^{- 13} (8 x + \frac{d}{d x} [- 14])$

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$- 12 {(4 x^{2} - 14)}^{- 13} (8 x + 0)$

Simplify the expression.

Add $8 x$ and $0$ .

$- 12 {(4 x^{2} - 14)}^{- 13} (8 x)$

Multiply $8$ by $- 12$ .

$- 96 {(4 x^{2} - 14)}^{- 13} x$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 96 \frac{1}{{(4 x^{2} - 14)}^{13}} x$

Combine terms.

Combine $- 96$ and $\frac{1}{{(4 x^{2} - 14)}^{13}}$ .

$\frac{- 96}{{(4 x^{2} - 14)}^{13}} x$

Move the negative in front of the fraction.

$- \frac{96}{{(4 x^{2} - 14)}^{13}} x$

Combine $x$ and $\frac{96}{{(4 x^{2} - 14)}^{13}}$ .

$- \frac{x \cdot 96}{{(4 x^{2} - 14)}^{13}}$

Move $96$ to the left of $x$ .

$- \frac{96 x}{{(4 x^{2} - 14)}^{13}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{96 x}{{(4 x^{2} - 14)}^{13}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{96 x}{{(4 x^{2} - 14)}^{13}}$

Find dy/dx y=(4x^2-14)^-12

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