Calculus Examples

$y = \frac{1}{2^{4 x}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{2^{4 x}})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Apply basic rules of exponents.

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Multiply $- 1$ by $4$ .

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $- 4 x$ .

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

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $2$ .

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

Step 3.2.3

Replace all occurrences of $u$ with $- 4 x$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

$2^{- 4 x} \ln (2) (- 4 \cdot 1)$

Step 3.3.3

Simplify with factoring out.

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Step 3.3.3.1

Multiply $- 4$ by $1$ .

$2^{- 4 x} \ln (2) \cdot - 4$

Step 3.3.3.2

Move $- 4$ to the left of $2^{- 4 x} \ln (2)$ .

$- 4 \cdot (2^{- 4 x} \ln (2))$

Step 3.3.3.3

Factor out negative.

$- (4 \cdot 2^{- 4 x}) \ln (2)$

Step 3.3.3.4

Rewrite $4$ as $2^{2}$ .

$- (2^{2} \cdot 2^{- 4 x}) \ln (2)$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2^{2 - 4 x} \ln (2)$

Step 4

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

$y' = - 2^{2 - 4 x} \ln (2)$

Step 5

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

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