Calculus Examples

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

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $4$ by $1$ .

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

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 $0$ .

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

Step 4.2

Add $4$ and $0$ .

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

Step 4.3

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Multiply $2$ by $5$ .

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

Step 6

Differentiate using the Constant Rule.

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Step 6.1

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

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

Step 6.2

Add $10 x$ and $0$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $4$ by $5$ .

$20 x^{2} - 1 \cdot 4 + 4 x (10 x) + 3 (10 x)$

Step 7.3.2

Multiply $- 1$ by $4$ .

$20 x^{2} - 4 + 4 x (10 x) + 3 (10 x)$

Step 7.3.3

Multiply $10$ by $4$ .

$20 x^{2} - 4 + 40 x \cdot x + 3 (10 x)$

Step 7.3.4

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

$20 x^{2} - 4 + 40 (x^{1} x) + 3 (10 x)$

Step 7.3.5

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

$20 x^{2} - 4 + 40 (x^{1} x^{1}) + 3 (10 x)$

Step 7.3.6

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

$20 x^{2} - 4 + 40 x^{1 + 1} + 3 (10 x)$

Step 7.3.7

Add $1$ and $1$ .

$20 x^{2} - 4 + 40 x^{2} + 3 (10 x)$

Step 7.3.8

Multiply $10$ by $3$ .

$20 x^{2} - 4 + 40 x^{2} + 30 x$

Step 7.3.9

Add $20 x^{2}$ and $40 x^{2}$ .

$60 x^{2} - 4 + 30 x$

Step 7.4

Reorder terms.

$60 x^{2} + 30 x - 4$