Find the Derivative - d/dx 1/((x^2+1)^4)

$\frac{1}{{(x^{2} + 1)}^{4}}$

Step 1

Apply basic rules of exponents.

Rewrite $\frac{1}{{(x^{2} + 1)}^{4}}$ as ${({(x^{2} + 1)}^{4})}^{- 1}$ .

$\frac{d}{d x} [{({(x^{2} + 1)}^{4})}^{- 1}]$

Multiply the exponents in ${({(x^{2} + 1)}^{4})}^{- 1}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [{(x^{2} + 1)}^{4 \cdot - 1}]$

Multiply $4$ by $- 1$ .

$\frac{d}{d x} [{(x^{2} + 1)}^{- 4}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 4}$ and $g (x) = x^{2} + 1$ .

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [x^{2} + 1]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 4$ .

$- 4 u^{- 5} \frac{d}{d x} [x^{2} + 1]$

Replace all occurrences of $u$ with $x^{2} + 1$ .

$- 4 {(x^{2} + 1)}^{- 5} \frac{d}{d x} [x^{2} + 1]$

Step 3

Differentiate.

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$- 4 {(x^{2} + 1)}^{- 5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])$

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

$- 4 {(x^{2} + 1)}^{- 5} (2 x + \frac{d}{d x} [1])$

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 4 {(x^{2} + 1)}^{- 5} (2 x + 0)$

Simplify the expression.

Add $2 x$ and $0$ .

$- 4 {(x^{2} + 1)}^{- 5} (2 x)$

Multiply $2$ by $- 4$ .

$- 8 {(x^{2} + 1)}^{- 5} x$

Step 4

Simplify.

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

$- 8 \frac{1}{{(x^{2} + 1)}^{5}} x$

Combine terms.

Combine $- 8$ and $\frac{1}{{(x^{2} + 1)}^{5}}$ .

$\frac{- 8}{{(x^{2} + 1)}^{5}} x$

Move the negative in front of the fraction.

$- \frac{8}{{(x^{2} + 1)}^{5}} x$

Combine $x$ and $\frac{8}{{(x^{2} + 1)}^{5}}$ .

$- \frac{x \cdot 8}{{(x^{2} + 1)}^{5}}$

Move $8$ to the left of $x$ .

$- \frac{8 x}{{(x^{2} + 1)}^{5}}$

Find the Derivative - d/dx 1/((x^2+1)^4)

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