Calculus Examples

$x^{5} + y^{5} \cdot (15 x y)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{5} + y^{5} \cdot (15 x y)$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [y^{5} \cdot (15 x y)]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [y^{5} \cdot (15 x y)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [y^{5} \cdot (15 x y)]$

Step 2

Evaluate $\frac{d}{d x} [y^{5} \cdot (15 x y)]$ .

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Step 2.1

Multiply $y^{5}$ by $y$ by adding the exponents.

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Step 2.1.1

Move $y$ .

$5 x^{4} + \frac{d}{d x} [y \cdot y^{5} \cdot (15 x)]$

Step 2.1.2

Multiply $y$ by $y^{5}$ .

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Step 2.1.2.1

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

$5 x^{4} + \frac{d}{d x} [y^{1} y^{5} \cdot (15 x)]$

Step 2.1.2.2

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

$5 x^{4} + \frac{d}{d x} [y^{1 + 5} \cdot (15 x)]$

Step 2.1.3

Add $1$ and $5$ .

$5 x^{4} + \frac{d}{d x} [y^{6} \cdot (15 x)]$

Step 2.2

Move $15$ to the left of $y^{6}$ .

$5 x^{4} + \frac{d}{d x} [15 \cdot y^{6} x]$

Step 2.3

Since $15 y^{6}$ is constant with respect to $x$ , the derivative of $15 y^{6} x$ with respect to $x$ is $15 y^{6} \frac{d}{d x} [x]$ .

$5 x^{4} + 15 y^{6} \frac{d}{d x} [x]$

Step 2.4

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

$5 x^{4} + 15 y^{6} \cdot 1$

Step 2.5

Multiply $15$ by $1$ .

$5 x^{4} + 15 y^{6}$

Step 3

Reorder terms.

$15 y^{6} + 5 x^{4}$