Calculus Examples

$y = (8 x + 5) (x^{2} - 3 x)$

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 $f (x) = 8 x + 5$ and $g (x) = x^{2} - 3 x$ .

$(8 x + 5) \frac{d}{d x} [x^{2} - 3 x] + (x^{2} - 3 x) \frac{d}{d x} [8 x + 5]$

Step 2

Differentiate.

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Step 2.1

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

$(8 x + 5) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]) + (x^{2} - 3 x) \frac{d}{d x} [8 x + 5]$

Step 2.2

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

$(8 x + 5) (2 x + \frac{d}{d x} [- 3 x]) + (x^{2} - 3 x) \frac{d}{d x} [8 x + 5]$

Step 2.3

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

$(8 x + 5) (2 x - 3 \frac{d}{d x} [x]) + (x^{2} - 3 x) \frac{d}{d x} [8 x + 5]$

Step 2.4

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

$(8 x + 5) (2 x - 3 \cdot 1) + (x^{2} - 3 x) \frac{d}{d x} [8 x + 5]$

Step 2.5

Multiply $- 3$ by $1$ .

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) \frac{d}{d x} [8 x + 5]$

Step 2.6

By the Sum Rule, the derivative of $8 x + 5$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [5]$ .

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) (\frac{d}{d x} [8 x] + \frac{d}{d x} [5])$

Step 2.7

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) (8 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 2.8

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

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) (8 \cdot 1 + \frac{d}{d x} [5])$

Step 2.9

Multiply $8$ by $1$ .

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) (8 + \frac{d}{d x} [5])$

Step 2.10

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

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) (8 + 0)$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $8$ and $0$ .

$(8 x + 5) (2 x - 3) + (x^{2} - 3 x) \cdot 8$

Step 2.11.2

Move $8$ to the left of $x^{2} - 3 x$ .

$(8 x + 5) (2 x - 3) + 8 \cdot (x^{2} - 3 x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(8 x + 5) (2 x - 3) + 8 x^{2} + 8 (- 3 x)$

Step 3.2

Apply the distributive property.

$(8 x + 5) (2 x) + (8 x + 5) \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.3

Apply the distributive property.

$8 x (2 x) + 5 (2 x) + (8 x + 5) \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.4

Apply the distributive property.

$8 x (2 x) + 5 (2 x) + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $2$ by $8$ .

$16 x \cdot x + 5 (2 x) + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.2

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

$16 (x^{1} x) + 5 (2 x) + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.3

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

$16 (x^{1} x^{1}) + 5 (2 x) + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.4

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

$16 x^{1 + 1} + 5 (2 x) + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.5

Add $1$ and $1$ .

$16 x^{2} + 5 (2 x) + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.6

Multiply $2$ by $5$ .

$16 x^{2} + 10 x + 8 x \cdot - 3 + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.7

Multiply $- 3$ by $8$ .

$16 x^{2} + 10 x - 24 x + 5 \cdot - 3 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.8

Multiply $5$ by $- 3$ .

$16 x^{2} + 10 x - 24 x - 15 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.9

Subtract $24 x$ from $10 x$ .

$16 x^{2} - 14 x - 15 + 8 x^{2} + 8 (- 3 x)$

Step 3.5.10

Multiply $- 3$ by $8$ .

$16 x^{2} - 14 x - 15 + 8 x^{2} - 24 x$

Step 3.5.11

Add $16 x^{2}$ and $8 x^{2}$ .

$24 x^{2} - 14 x - 15 - 24 x$

Step 3.5.12

Subtract $24 x$ from $- 14 x$ .

$24 x^{2} - 38 x - 15$