Calculus Examples

$y = x^{3} + 6 x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{3} + 6 x^{2})$

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

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $x^{3} + 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x^{2}]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x^{2}]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [6 x^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 3.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} + 6 \frac{d}{d x} [x^{2}]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 x^{2} + 6 (2 x)$

Step 3.2.3

Multiply $2$ by $6$ .

$3 x^{2} + 12 x$

Step 4

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

$y' = 3 x^{2} + 12 x$

Step 5

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

$\frac{d y}{d x} = 3 x^{2} + 12 x$