Calculus Examples

$h (x) = {(x^{2} + 5)}^{2} (x - 5)$

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 $f (x) = {(x^{2} + 5)}^{2}$ and $g (x) = x - 5$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

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

Step 2.2

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

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

Step 2.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

${(x^{2} + 5)}^{2} (1 + 0) + (x - 5) \frac{d}{d x} [{(x^{2} + 5)}^{2}]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply ${(x^{2} + 5)}^{2}$ by $1$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 5$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2} + 5$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x^{2} + 5$ .

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

Step 4

Differentiate.

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Step 4.1

Move $2$ to the left of $x - 5$ .

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

Step 4.2

By the Sum Rule, the derivative of $x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

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

Step 4.3

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

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

Step 4.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

${(x^{2} + 5)}^{2} + 2 (x - 5) (x^{2} + 5) (2 x + 0)$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $2 x$ and $0$ .

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

Step 4.5.2

Multiply $2$ by $2$ .

${(x^{2} + 5)}^{2} + 4 (x - 5) (x^{2} + 5) x$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

${(x^{2} + 5)}^{2} + (4 x + 4 \cdot - 5) (x^{2} + 5) x$

Step 5.2

Multiply $4$ by $- 5$ .

${(x^{2} + 5)}^{2} + (4 x - 20) (x^{2} + 5) x$

Step 5.3

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{2} + (4 x - 20) (x^{2} + 5) x$ .

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Step 5.3.1

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{2}$ .

$(x^{2} + 5) (x^{2} + 5) + (4 x - 20) (x^{2} + 5) x$

Step 5.3.2

Factor $x^{2} + 5$ out of $(4 x - 20) (x^{2} + 5) x$ .

$(x^{2} + 5) (x^{2} + 5) + (x^{2} + 5) ((4 x - 20) x)$

Step 5.3.3

Factor $x^{2} + 5$ out of $(x^{2} + 5) (x^{2} + 5) + (x^{2} + 5) ((4 x - 20) x)$ .

$(x^{2} + 5) (x^{2} + 5 + (4 x - 20) x)$

Step 5.4

Simplify each term.

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Step 5.4.1

Apply the distributive property.

$(x^{2} + 5) (x^{2} + 5 + 4 x \cdot x - 20 x)$

Step 5.4.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.2.1

Move $x$ .

$(x^{2} + 5) (x^{2} + 5 + 4 (x \cdot x) - 20 x)$

Step 5.4.2.2

Multiply $x$ by $x$ .

$(x^{2} + 5) (x^{2} + 5 + 4 x^{2} - 20 x)$

Step 5.5

Add $x^{2}$ and $4 x^{2}$ .

$(x^{2} + 5) (5 x^{2} + 5 - 20 x)$

Step 5.6

Expand $(x^{2} + 5) (5 x^{2} + 5 - 20 x)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} (5 x^{2}) + x^{2} \cdot 5 + x^{2} (- 20 x) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7

Simplify each term.

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Step 5.7.1

Rewrite using the commutative property of multiplication.

$5 x^{2} x^{2} + x^{2} \cdot 5 + x^{2} (- 20 x) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.7.2.1

Move $x^{2}$ .

$5 (x^{2} x^{2}) + x^{2} \cdot 5 + x^{2} (- 20 x) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.2.2

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

$5 x^{2 + 2} + x^{2} \cdot 5 + x^{2} (- 20 x) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.2.3

Add $2$ and $2$ .

$5 x^{4} + x^{2} \cdot 5 + x^{2} (- 20 x) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.3

Move $5$ to the left of $x^{2}$ .

$5 x^{4} + 5 \cdot x^{2} + x^{2} (- 20 x) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.4

Rewrite using the commutative property of multiplication.

$5 x^{4} + 5 x^{2} - 20 x^{2} x + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.7.5.1

Move $x$ .

$5 x^{4} + 5 x^{2} - 20 (x \cdot x^{2}) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.5.2

Multiply $x$ by $x^{2}$ .

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Step 5.7.5.2.1

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

$5 x^{4} + 5 x^{2} - 20 (x^{1} x^{2}) + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.5.2.2

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

$5 x^{4} + 5 x^{2} - 20 x^{1 + 2} + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.5.3

Add $1$ and $2$ .

$5 x^{4} + 5 x^{2} - 20 x^{3} + 5 (5 x^{2}) + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.6

Multiply $5$ by $5$ .

$5 x^{4} + 5 x^{2} - 20 x^{3} + 25 x^{2} + 5 \cdot 5 + 5 (- 20 x)$

Step 5.7.7

Multiply $5$ by $5$ .

$5 x^{4} + 5 x^{2} - 20 x^{3} + 25 x^{2} + 25 + 5 (- 20 x)$

Step 5.7.8

Multiply $- 20$ by $5$ .

$5 x^{4} + 5 x^{2} - 20 x^{3} + 25 x^{2} + 25 - 100 x$

Step 5.8

Add $5 x^{2}$ and $25 x^{2}$ .

$5 x^{4} - 20 x^{3} + 30 x^{2} + 25 - 100 x$