Calculus Examples

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

Step 1

Factor $2$ out of $2 x$ .

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

Step 2

Apply the product rule to $2 (x)$ .

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

Step 3

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

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

Step 4

Raise $2$ to the power of $5$ .

$\frac{d}{d x} [\frac{4}{\frac{1}{32} x^{- 5}}]$

Step 5

Move $x^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{d}{d x} [\frac{4 x^{5}}{\frac{1}{32}}]$

Step 6

Multiply by the reciprocal of the fraction to divide by $\frac{1}{32}$ .

$\frac{d}{d x} [4 x^{5} \cdot 32]$

Step 7

Multiply $32$ by $4$ .

$\frac{d}{d x} [128 x^{5}]$

Step 8

Since $128$ is constant with respect to $x$ , the derivative of $128 x^{5}$ with respect to $x$ is $128 \frac{d}{d x} [x^{5}]$ .

$128 \frac{d}{d x} [x^{5}]$

Step 9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$128 (5 x^{4})$

Step 10

Multiply $5$ by $128$ .

$640 x^{4}$