Calculus Examples

$f (x) = x^{5} h (x)$

Step 1

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 $f (x) = x^{5} h$ and $g (x) = x$ .

$x^{5} h \frac{d}{d x} [x] + x \frac{d}{d x} [x^{5} h]$

Step 2

Differentiate.

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Step 2.1

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

$x^{5} h \cdot 1 + x \frac{d}{d x} [x^{5} h]$

Step 2.2

Since $h$ is constant with respect to $x$ , the derivative of $x^{5} h$ with respect to $x$ is $h \frac{d}{d x} [x^{5}]$ .

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

Step 2.3

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

$x^{5} h \cdot 1 + x (h (5 x^{4}))$

Step 2.4

Reorder.

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Step 2.4.1

Move $5$ to the left of $h$ .

$x^{5} h \cdot 1 + x (5 \cdot h x^{4})$

Step 2.4.2

Reorder the factors of $5 h x^{4}$ .

$x^{5} h \cdot 1 + x (5 x^{4} h)$

Step 3

Simplify.

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Step 3.1

Combine terms.

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Step 3.1.1

Multiply $x^{5}$ by $1$ .

$h \cdot x^{5} + x (5 x^{4} h)$

Step 3.1.2

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

$h x^{5} + 5 (x^{1} x^{4}) h$

Step 3.1.3

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

$h x^{5} + 5 x^{1 + 4} h$

Step 3.1.4

Add $1$ and $4$ .

$h x^{5} + 5 x^{5} h$

Step 3.1.5

Add $h x^{5}$ and $5 x^{5} h$ .

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Step 3.1.5.1

Move $x^{5}$ .

$h x^{5} + 5 h x^{5}$

Step 3.1.5.2

Add $h x^{5}$ and $5 h x^{5}$ .

$6 h x^{5}$

Step 3.2

Reorder the factors of $6 h x^{5}$ .

$6 x^{5} h$