Precalculus Examples

$f (x) = (5 x^{5} + 5) (- 2 x^{5} - 3)$

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 $f (x) = 5 x^{5} + 5$ and $g (x) = - 2 x^{5} - 3$ .

$(5 x^{5} + 5) \frac{d}{d x} [- 2 x^{5} - 3] + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 2

By the Sum Rule, the derivative of $- 2 x^{5} - 3$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{5}] + \frac{d}{d x} [- 3]$ .

$(5 x^{5} + 5) (\frac{d}{d x} [- 2 x^{5}] + \frac{d}{d x} [- 3]) + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 3

Evaluate $\frac{d}{d x} [- 2 x^{5}]$ .

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{5}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{5}]$ .

$(5 x^{5} + 5) (- 2 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 3]) + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 3.2

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

$(5 x^{5} + 5) (- 2 (5 x^{4}) + \frac{d}{d x} [- 3]) + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 3.3

Multiply $5$ by $- 2$ .

$(5 x^{5} + 5) (- 10 x^{4} + \frac{d}{d x} [- 3]) + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 4

Differentiate.

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Step 4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(5 x^{5} + 5) (- 10 x^{4} + 0) + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 4.2

Add $- 10 x^{4}$ and $0$ .

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) \frac{d}{d x} [5 x^{5} + 5]$

Step 4.3

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

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) (\frac{d}{d x} [5 x^{5}] + \frac{d}{d x} [5])$

Step 5

Evaluate $\frac{d}{d x} [5 x^{5}]$ .

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Step 5.1

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

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) (5 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [5])$

Step 5.2

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

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) (5 (5 x^{4}) + \frac{d}{d x} [5])$

Step 5.3

Multiply $5$ by $5$ .

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) (25 x^{4} + \frac{d}{d x} [5])$

Step 6

Differentiate using the Constant Rule.

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Step 6.1

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

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) (25 x^{4} + 0)$

Step 6.2

Add $25 x^{4}$ and $0$ .

$(5 x^{5} + 5) (- 10 x^{4}) + (- 2 x^{5} - 3) (25 x^{4})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$5 x^{5} (- 10 x^{4}) + 5 (- 10 x^{4}) + (- 2 x^{5} - 3) (25 x^{4})$

Step 7.2

Apply the distributive property.

$5 x^{5} (- 10 x^{4}) + 5 (- 10 x^{4}) - 2 x^{5} (25 x^{4}) - 3 (25 x^{4})$

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $- 10$ by $5$ .

$- 50 x^{5} x^{4} + 5 (- 10 x^{4}) - 2 x^{5} (25 x^{4}) - 3 (25 x^{4})$

Step 7.3.2

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 7.3.2.1

Move $x^{4}$ .

$- 50 (x^{4} x^{5}) + 5 (- 10 x^{4}) - 2 x^{5} (25 x^{4}) - 3 (25 x^{4})$

Step 7.3.2.2

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

$- 50 x^{4 + 5} + 5 (- 10 x^{4}) - 2 x^{5} (25 x^{4}) - 3 (25 x^{4})$

Step 7.3.2.3

Add $4$ and $5$ .

$- 50 x^{9} + 5 (- 10 x^{4}) - 2 x^{5} (25 x^{4}) - 3 (25 x^{4})$

Step 7.3.3

Multiply $- 10$ by $5$ .

$- 50 x^{9} - 50 x^{4} - 2 x^{5} (25 x^{4}) - 3 (25 x^{4})$

Step 7.3.4

Multiply $25$ by $- 2$ .

$- 50 x^{9} - 50 x^{4} - 50 x^{5} x^{4} - 3 (25 x^{4})$

Step 7.3.5

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 7.3.5.1

Move $x^{4}$ .

$- 50 x^{9} - 50 x^{4} - 50 (x^{4} x^{5}) - 3 (25 x^{4})$

Step 7.3.5.2

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

$- 50 x^{9} - 50 x^{4} - 50 x^{4 + 5} - 3 (25 x^{4})$

Step 7.3.5.3

Add $4$ and $5$ .

$- 50 x^{9} - 50 x^{4} - 50 x^{9} - 3 (25 x^{4})$

Step 7.3.6

Multiply $25$ by $- 3$ .

$- 50 x^{9} - 50 x^{4} - 50 x^{9} - 75 x^{4}$

Step 7.3.7

Subtract $50 x^{9}$ from $- 50 x^{9}$ .

$- 100 x^{9} - 50 x^{4} - 75 x^{4}$

Step 7.3.8

Subtract $75 x^{4}$ from $- 50 x^{4}$ .

$- 100 x^{9} - 125 x^{4}$