Calculus Examples

$y = (3 x^{4} - 7) \cdot (5 x^{2} + 1)$

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 $f (x) = 3 x^{4} - 7$ and $g (x) = 5 x^{2} + 1$ .

$(3 x^{4} - 7) \frac{d}{d x} [5 x^{2} + 1] + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [1]$ .

$(3 x^{4} - 7) (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [1]) + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.2

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

$(3 x^{4} - 7) (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.3

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

$(3 x^{4} - 7) (5 (2 x) + \frac{d}{d x} [1]) + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.4

Multiply $2$ by $5$ .

$(3 x^{4} - 7) (10 x + \frac{d}{d x} [1]) + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(3 x^{4} - 7) (10 x + 0) + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $10 x$ and $0$ .

$(3 x^{4} - 7) (10 x) + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.6.2

Move $10$ to the left of $3 x^{4} - 7$ .

$10 \cdot (3 x^{4} - 7) x + (5 x^{2} + 1) \frac{d}{d x} [3 x^{4} - 7]$

Step 2.7

By the Sum Rule, the derivative of $3 x^{4} - 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 7]$ .

$10 (3 x^{4} - 7) x + (5 x^{2} + 1) (\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 7])$

Step 2.8

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$10 (3 x^{4} - 7) x + (5 x^{2} + 1) (3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 7])$

Step 2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$10 (3 x^{4} - 7) x + (5 x^{2} + 1) (3 (4 x^{3}) + \frac{d}{d x} [- 7])$

Step 2.10

Multiply $4$ by $3$ .

$10 (3 x^{4} - 7) x + (5 x^{2} + 1) (12 x^{3} + \frac{d}{d x} [- 7])$

Step 2.11

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$10 (3 x^{4} - 7) x + (5 x^{2} + 1) (12 x^{3} + 0)$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $12 x^{3}$ and $0$ .

$10 (3 x^{4} - 7) x + (5 x^{2} + 1) (12 x^{3})$

Step 2.12.2

Move $12$ to the left of $5 x^{2} + 1$ .

$10 (3 x^{4} - 7) x + 12 \cdot (5 x^{2} + 1) x^{3}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(10 (3 x^{4}) + 10 \cdot - 7) x + 12 (5 x^{2} + 1) x^{3}$

Step 3.2

Apply the distributive property.

$10 (3 x^{4}) x + 10 \cdot - 7 x + 12 (5 x^{2} + 1) x^{3}$

Step 3.3

Apply the distributive property.

$10 (3 x^{4}) x + 10 \cdot - 7 x + (12 (5 x^{2}) + 12 \cdot 1) x^{3}$

Step 3.4

Apply the distributive property.

$10 (3 x^{4}) x + 10 \cdot - 7 x + 12 (5 x^{2}) x^{3} + 12 \cdot 1 x^{3}$

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $3$ by $10$ .

$30 x^{4} x + 10 \cdot - 7 x + 12 (5 x^{2}) x^{3} + 12 \cdot 1 x^{3}$

Step 3.5.2

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

$30 (x^{1} x^{4}) + 10 \cdot - 7 x + 12 (5 x^{2}) x^{3} + 12 \cdot 1 x^{3}$

Step 3.5.3

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

$30 x^{1 + 4} + 10 \cdot - 7 x + 12 (5 x^{2}) x^{3} + 12 \cdot 1 x^{3}$

Step 3.5.4

Add $1$ and $4$ .

$30 x^{5} + 10 \cdot - 7 x + 12 (5 x^{2}) x^{3} + 12 \cdot 1 x^{3}$

Step 3.5.5

Multiply $10$ by $- 7$ .

$30 x^{5} - 70 x + 12 (5 x^{2}) x^{3} + 12 \cdot 1 x^{3}$

Step 3.5.6

Multiply $5$ by $12$ .

$30 x^{5} - 70 x + 60 x^{2} x^{3} + 12 \cdot 1 x^{3}$

Step 3.5.7

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.5.7.1

Move $x^{3}$ .

$30 x^{5} - 70 x + 60 (x^{3} x^{2}) + 12 \cdot 1 x^{3}$

Step 3.5.7.2

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

$30 x^{5} - 70 x + 60 x^{3 + 2} + 12 \cdot 1 x^{3}$

Step 3.5.7.3

Add $3$ and $2$ .

$30 x^{5} - 70 x + 60 x^{5} + 12 \cdot 1 x^{3}$

Step 3.5.8

Multiply $12$ by $1$ .

$30 x^{5} - 70 x + 60 x^{5} + 12 x^{3}$

Step 3.5.9

Add $30 x^{5}$ and $60 x^{5}$ .

$90 x^{5} - 70 x + 12 x^{3}$

Step 3.6

Reorder terms.

$90 x^{5} + 12 x^{3} - 70 x$