Calculus Examples

$y = \frac{7^{x}}{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{7^{x}}{2})$

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

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{7^{x}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [7^{x}]$ .

$\frac{1}{2} \frac{d}{d x} [7^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7$ .

$\frac{1}{2} (7^{x} \ln (7))$

Step 3.3

Combine fractions.

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Step 3.3.1

Combine $7^{x}$ and $\frac{1}{2}$ .

$\frac{7^{x}}{2} \ln (7)$

Step 3.3.2

Combine $\frac{7^{x}}{2}$ and $\ln (7)$ .

$\frac{7^{x} \ln (7)}{2}$

Step 4

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

$y' = \frac{7^{x} \ln (7)}{2}$

Step 5

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

$\frac{d y}{d x} = \frac{7^{x} \ln (7)}{2}$