Calculus Examples

$y = 5 x (x^{2} + 6)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 x (x^{2} + 6))$

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 $5$ is constant with respect to $x$ , the derivative of $5 x (x^{2} + 6)$ with respect to $x$ is $5 \frac{d}{d x} [x (x^{2} + 6)]$ .

$5 \frac{d}{d x} [x (x^{2} + 6)]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} + 6$ .

$5 (x \frac{d}{d x} [x^{2} + 6] + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.3

Differentiate.

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Step 3.3.1

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

$5 (x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6]) + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.3.2

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

$5 (x (2 x + \frac{d}{d x} [6]) + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.3.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$5 (x (2 x + 0) + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.3.4

Add $2 x$ and $0$ .

$5 (x (2 x) + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.4

Raise $x$ to the power of $1$ .

$5 (2 (x^{1} x) + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.5

Raise $x$ to the power of $1$ .

$5 (2 (x^{1} x^{1}) + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.6

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

$5 (2 x^{1 + 1} + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.7

Add $1$ and $1$ .

$5 (2 x^{2} + (x^{2} + 6) \frac{d}{d x} [x])$

Step 3.8

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

$5 (2 x^{2} + (x^{2} + 6) \cdot 1)$

Step 3.9

Simplify by adding terms.

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Step 3.9.1

Multiply $x^{2} + 6$ by $1$ .

$5 (2 x^{2} + x^{2} + 6)$

Step 3.9.2

Add $2 x^{2}$ and $x^{2}$ .

$5 (3 x^{2} + 6)$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$5 (3 x^{2}) + 5 \cdot 6$

Step 3.10.2

Combine terms.

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Step 3.10.2.1

Multiply $3$ by $5$ .

$15 x^{2} + 5 \cdot 6$

Step 3.10.2.2

Multiply $5$ by $6$ .

$15 x^{2} + 30$

Step 4

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

$y' = 15 x^{2} + 30$

Step 5

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

$\frac{d y}{d x} = 15 x^{2} + 30$