Find dy/dx y=(3x^2-9)^-11

$y = {(3 x^{2} - 9)}^{- 11}$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(3 x^{2} - 9)}^{- 11})$

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

$y'$

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 $f (x) = x^{- 11}$ and $g (x) = 3 x^{2} - 9$ .

To apply the Chain Rule, set $u$ as $3 x^{2} - 9$ .

$\frac{d}{d u} [u^{- 11}] \frac{d}{d x} [3 x^{2} - 9]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 11$ .

$- 11 u^{- 12} \frac{d}{d x} [3 x^{2} - 9]$

Replace all occurrences of $u$ with $3 x^{2} - 9$ .

$- 11 {(3 x^{2} - 9)}^{- 12} \frac{d}{d x} [3 x^{2} - 9]$

Differentiate.

By the Sum Rule, the derivative of $3 x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 9]$ .

$- 11 {(3 x^{2} - 9)}^{- 12} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 9])$

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$- 11 {(3 x^{2} - 9)}^{- 12} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9])$

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

$- 11 {(3 x^{2} - 9)}^{- 12} (3 (2 x) + \frac{d}{d x} [- 9])$

Multiply $2$ by $3$ .

$- 11 {(3 x^{2} - 9)}^{- 12} (6 x + \frac{d}{d x} [- 9])$

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$- 11 {(3 x^{2} - 9)}^{- 12} (6 x + 0)$

Simplify the expression.

Add $6 x$ and $0$ .

$- 11 {(3 x^{2} - 9)}^{- 12} (6 x)$

Multiply $6$ by $- 11$ .

$- 66 {(3 x^{2} - 9)}^{- 12} x$

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

$- 66 \frac{1}{{(3 x^{2} - 9)}^{12}} x$

Combine terms.

Combine $- 66$ and $\frac{1}{{(3 x^{2} - 9)}^{12}}$ .

$\frac{- 66}{{(3 x^{2} - 9)}^{12}} x$

Move the negative in front of the fraction.

$- \frac{66}{{(3 x^{2} - 9)}^{12}} x$

Combine $x$ and $\frac{66}{{(3 x^{2} - 9)}^{12}}$ .

$- \frac{x \cdot 66}{{(3 x^{2} - 9)}^{12}}$

Move $66$ to the left of $x$ .

$- \frac{66 x}{{(3 x^{2} - 9)}^{12}}$

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

$y' = - \frac{66 x}{{(3 x^{2} - 9)}^{12}}$

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

$\frac{d y}{d x} = - \frac{66 x}{{(3 x^{2} - 9)}^{12}}$

Find dy/dx y=(3x^2-9)^-11

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