Calculus Examples

$y = 3^{2 x}$

Step 1

Differentiate both sides of the equation.

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

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 3^{x}$ and $g (x) = 2 x$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $3$ .

$3^{u} \ln (3) \frac{d}{d x} [2 x]$

Step 3.1.3

Replace all occurrences of $u$ with $2 x$ .

$3^{2 x} \ln (3) \frac{d}{d x} [2 x]$

Step 3.2

Differentiate.

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Step 3.2.1

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

$3^{2 x} \ln (3) (2 \frac{d}{d x} [x])$

Step 3.2.2

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

$3^{2 x} \ln (3) (2 \cdot 1)$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $2$ by $1$ .

$3^{2 x} \ln (3) \cdot 2$

Step 3.2.3.2

Move $2$ to the left of $3^{2 x} \ln (3)$ .

$2 \cdot 3^{2 x} \ln (3)$

Step 4

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

$y' = 2 \cdot (3^{2 x} \ln (3))$

Step 5

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

$\frac{d y}{d x} = 2 \cdot (3^{2 x} \ln (3))$