Calculus Examples

$y = (3 - 3 x^{2}) (5 x^{2} - 60)$

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 $f (x) = 3 - 3 x^{2}$ and $g (x) = 5 x^{2} - 60$ .

$(3 - 3 x^{2}) \frac{d}{d x} [5 x^{2} - 60] + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{2} - 60$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 60]$ .

$(3 - 3 x^{2}) (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 60]) + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.2

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

$(3 - 3 x^{2}) (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 60]) + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.3

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

$(3 - 3 x^{2}) (5 (2 x) + \frac{d}{d x} [- 60]) + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.4

Multiply $2$ by $5$ .

$(3 - 3 x^{2}) (10 x + \frac{d}{d x} [- 60]) + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.5

Since $- 60$ is constant with respect to $x$ , the derivative of $- 60$ with respect to $x$ is $0$ .

$(3 - 3 x^{2}) (10 x + 0) + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $10 x$ and $0$ .

$(3 - 3 x^{2}) (10 x) + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.6.2

Move $10$ to the left of $3 - 3 x^{2}$ .

$10 \cdot (3 - 3 x^{2}) x + (5 x^{2} - 60) \frac{d}{d x} [3 - 3 x^{2}]$

Step 2.7

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

$10 (3 - 3 x^{2}) x + (5 x^{2} - 60) (\frac{d}{d x} [3] + \frac{d}{d x} [- 3 x^{2}])$

Step 2.8

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

$10 (3 - 3 x^{2}) x + (5 x^{2} - 60) (0 + \frac{d}{d x} [- 3 x^{2}])$

Step 2.9

Add $0$ and $\frac{d}{d x} [- 3 x^{2}]$ .

$10 (3 - 3 x^{2}) x + (5 x^{2} - 60) \frac{d}{d x} [- 3 x^{2}]$

Step 2.10

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$10 (3 - 3 x^{2}) x + (5 x^{2} - 60) (- 3 \frac{d}{d x} [x^{2}])$

Step 2.11

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

$10 (3 - 3 x^{2}) x + (5 x^{2} - 60) (- 3 (2 x))$

Step 2.12

Multiply $2$ by $- 3$ .

$10 (3 - 3 x^{2}) x + (5 x^{2} - 60) (- 6 x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(10 \cdot 3 + 10 (- 3 x^{2})) x + (5 x^{2} - 60) (- 6 x)$

Step 3.2

Apply the distributive property.

$10 \cdot 3 x + 10 (- 3 x^{2}) x + (5 x^{2} - 60) (- 6 x)$

Step 3.3

Apply the distributive property.

$10 \cdot 3 x + 10 (- 3 x^{2}) x + 5 x^{2} (- 6 x) - 60 (- 6 x)$

Step 3.4

Combine terms.

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Step 3.4.1

Multiply $10$ by $3$ .

$30 x + 10 (- 3 x^{2}) x + 5 x^{2} (- 6 x) - 60 (- 6 x)$

Step 3.4.2

Multiply $- 3$ by $10$ .

$30 x - 30 x^{2} x + 5 x^{2} (- 6 x) - 60 (- 6 x)$

Step 3.4.3

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

$30 x - 30 (x^{1} x^{2}) + 5 x^{2} (- 6 x) - 60 (- 6 x)$

Step 3.4.4

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

$30 x - 30 x^{1 + 2} + 5 x^{2} (- 6 x) - 60 (- 6 x)$

Step 3.4.5

Add $1$ and $2$ .

$30 x - 30 x^{3} + 5 x^{2} (- 6 x) - 60 (- 6 x)$

Step 3.4.6

Multiply $- 6$ by $5$ .

$30 x - 30 x^{3} - 30 x^{2} x - 60 (- 6 x)$

Step 3.4.7

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

$30 x - 30 x^{3} - 30 (x^{1} x^{2}) - 60 (- 6 x)$

Step 3.4.8

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

$30 x - 30 x^{3} - 30 x^{1 + 2} - 60 (- 6 x)$

Step 3.4.9

Add $1$ and $2$ .

$30 x - 30 x^{3} - 30 x^{3} - 60 (- 6 x)$

Step 3.4.10

Multiply $- 6$ by $- 60$ .

$30 x - 30 x^{3} - 30 x^{3} + 360 x$

Step 3.4.11

Subtract $30 x^{3}$ from $- 30 x^{3}$ .

$30 x - 60 x^{3} + 360 x$

Step 3.4.12

Add $30 x$ and $360 x$ .

$- 60 x^{3} + 390 x$