Calculus Examples

$(h^{3} - 5) (3 h^{2} - 5 h - 4)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d h} [f (h) g (h)]$ is $f (h) \frac{d}{d h} [g (h)] + g (h) \frac{d}{d h} [f (h)]$ where $f (h) = h^{3} - 5$ and $g (h) = 3 h^{2} - 5 h - 4$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 h^{2} - 5 h - 4$ with respect to $h$ is $\frac{d}{d h} [3 h^{2}] + \frac{d}{d h} [- 5 h] + \frac{d}{d h} [- 4]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $3$ .

$(h^{3} - 5) (6 h + \frac{d}{d h} [- 5 h] + \frac{d}{d h} [- 4]) + (3 h^{2} - 5 h - 4) \frac{d}{d h} [h^{3} - 5]$

Step 2.5

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

$(h^{3} - 5) (6 h - 5 \frac{d}{d h} [h] + \frac{d}{d h} [- 4]) + (3 h^{2} - 5 h - 4) \frac{d}{d h} [h^{3} - 5]$

Step 2.6

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

$(h^{3} - 5) (6 h - 5 \cdot 1 + \frac{d}{d h} [- 4]) + (3 h^{2} - 5 h - 4) \frac{d}{d h} [h^{3} - 5]$

Step 2.7

Multiply $- 5$ by $1$ .

$(h^{3} - 5) (6 h - 5 + \frac{d}{d h} [- 4]) + (3 h^{2} - 5 h - 4) \frac{d}{d h} [h^{3} - 5]$

Step 2.8

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

$(h^{3} - 5) (6 h - 5 + 0) + (3 h^{2} - 5 h - 4) \frac{d}{d h} [h^{3} - 5]$

Step 2.9

Add $6 h - 5$ and $0$ .

$(h^{3} - 5) (6 h - 5) + (3 h^{2} - 5 h - 4) \frac{d}{d h} [h^{3} - 5]$

Step 2.10

By the Sum Rule, the derivative of $h^{3} - 5$ with respect to $h$ is $\frac{d}{d h} [h^{3}] + \frac{d}{d h} [- 5]$ .

$(h^{3} - 5) (6 h - 5) + (3 h^{2} - 5 h - 4) (\frac{d}{d h} [h^{3}] + \frac{d}{d h} [- 5])$

Step 2.11

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 3$ .

$(h^{3} - 5) (6 h - 5) + (3 h^{2} - 5 h - 4) (3 h^{2} + \frac{d}{d h} [- 5])$

Step 2.12

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

$(h^{3} - 5) (6 h - 5) + (3 h^{2} - 5 h - 4) (3 h^{2} + 0)$

Step 2.13

Simplify the expression.

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Step 2.13.1

Add $3 h^{2}$ and $0$ .

$(h^{3} - 5) (6 h - 5) + (3 h^{2} - 5 h - 4) (3 h^{2})$

Step 2.13.2

Move $3$ to the left of $3 h^{2} - 5 h - 4$ .

$(h^{3} - 5) (6 h - 5) + 3 \cdot (3 h^{2} - 5 h - 4) h^{2}$

Step 3

Simplify.

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Step 3.1