Find dy/dx y=2/(2x^4-5)

$y = \frac{2}{2 x^{4} - 5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{2}{2 x^{4} - 5})$

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 Constant Multiple Rule.

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{2 x^{4} - 5}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{2 x^{4} - 5}]$ .

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

Rewrite $\frac{1}{2 x^{4} - 5}$ as ${(2 x^{4} - 5)}^{- 1}$ .

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 2 x^{4} - 5$ .

To apply the Chain Rule, set $u$ as $2 x^{4} - 5$ .

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

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

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

Replace all occurrences of $u$ with $2 x^{4} - 5$ .

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

Differentiate.

Multiply $- 1$ by $2$ .

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

By the Sum Rule, the derivative of $2 x^{4} - 5$ with respect to $x$ is $\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [- 5]$ .

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

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

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

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

$- 2 {(2 x^{4} - 5)}^{- 2} (2 (4 x^{3}) + \frac{d}{d x} [- 5])$

Multiply $4$ by $2$ .

$- 2 {(2 x^{4} - 5)}^{- 2} (8 x^{3} + \frac{d}{d x} [- 5])$

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$- 2 {(2 x^{4} - 5)}^{- 2} (8 x^{3} + 0)$

Simplify the expression.

Add $8 x^{3}$ and $0$ .

$- 2 {(2 x^{4} - 5)}^{- 2} (8 x^{3})$

Multiply $8$ by $- 2$ .

$- 16 {(2 x^{4} - 5)}^{- 2} x^{3}$

Simplify.

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

$- 16 \frac{1}{{(2 x^{4} - 5)}^{2}} x^{3}$

Combine terms.

Combine $- 16$ and $\frac{1}{{(2 x^{4} - 5)}^{2}}$ .

$\frac{- 16}{{(2 x^{4} - 5)}^{2}} x^{3}$

Move the negative in front of the fraction.

$- \frac{16}{{(2 x^{4} - 5)}^{2}} x^{3}$

Combine $x^{3}$ and $\frac{16}{{(2 x^{4} - 5)}^{2}}$ .

$- \frac{x^{3} \cdot 16}{{(2 x^{4} - 5)}^{2}}$

Move $16$ to the left of $x^{3}$ .

$- \frac{16 x^{3}}{{(2 x^{4} - 5)}^{2}}$

Step 4

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

$y' = - \frac{16 x^{3}}{{(2 x^{4} - 5)}^{2}}$

Step 5

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

$\frac{d y}{d x} = - \frac{16 x^{3}}{{(2 x^{4} - 5)}^{2}}$

Find dy/dx y=2/(2x^4-5)

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