Calculus Examples

$f (x) = (x - 4) (4 x + 5)$

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 $f (x) = x - 4$ and $g (x) = 4 x + 5$ .

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

Step 2

By the Sum Rule, the derivative of $4 x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [5]$ .

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

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

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

Step 3.2

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

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

Step 3.3

Multiply $4$ by $1$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Add $4$ and $0$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$(x - 4) \cdot 4 + (4 x + 5) (1 + 0)$

Step 4.6

Add $1$ and $0$ .

$(x - 4) \cdot 4 + (4 x + 5) \cdot 1$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$x \cdot 4 - 4 \cdot 4 + (4 x + 5) \cdot 1$

Step 5.2

Apply the distributive property.

$x \cdot 4 - 4 \cdot 4 + 4 x \cdot 1 + 5 \cdot 1$

Step 5.3

Combine terms.

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Step 5.3.1

Move $4$ to the left of $x$ .

$4 \cdot x - 4 \cdot 4 + 4 x \cdot 1 + 5 \cdot 1$

Step 5.3.2

Multiply $- 4$ by $4$ .

$4 x - 16 + 4 x \cdot 1 + 5 \cdot 1$

Step 5.3.3

Multiply $4$ by $1$ .

$4 x - 16 + 4 x + 5 \cdot 1$

Step 5.3.4

Multiply $5$ by $1$ .

$4 x - 16 + 4 x + 5$

Step 5.3.5

Add $4 x$ and $4 x$ .

$8 x - 16 + 5$

Step 5.3.6

Add $- 16$ and $5$ .

$8 x - 11$