Calculus Examples

$s = (t^{4} - 3 t^{3} + 3) (1 - 2 t^{3})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d s} [f (s) g (s)]$ is $f (s) \frac{d}{d s} [g (s)] + g (s) \frac{d}{d s} [f (s)]$ where $f (s) = t^{4} - 3 t^{3} + 3$ and $g (s) = 1 - 2 t^{3}$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 - 2 t^{3}$ with respect to $s$ is $\frac{d}{d s} [1] + \frac{d}{d s} [- 2 t^{3}]$ .

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

Step 2.2

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

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

Step 2.3

Since $- 2 t^{3}$ is constant with respect to $s$ , the derivative of $- 2 t^{3}$ with respect to $s$ is $0$ .

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

Step 2.4

Add $0$ and $0$ .

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) \frac{d}{d s} [t^{4} - 3 t^{3} + 3]$

Step 2.5

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

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) (\frac{d}{d s} [t^{4}] + \frac{d}{d s} [- 3 t^{3}] + \frac{d}{d s} [3])$

Step 2.6

Since $t^{4}$ is constant with respect to $s$ , the derivative of $t^{4}$ with respect to $s$ is $0$ .

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) (0 + \frac{d}{d s} [- 3 t^{3}] + \frac{d}{d s} [3])$

Step 2.7

Since $- 3 t^{3}$ is constant with respect to $s$ , the derivative of $- 3 t^{3}$ with respect to $s$ is $0$ .

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) (0 + 0 + \frac{d}{d s} [3])$

Step 2.8

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

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) (0 + 0 + 0)$

Step 3

Combine terms.

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Step 3.1

Add $0$ and $0$ .

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) (0 + 0)$

Step 3.2

Add $0$ and $0$ .

$(t^{4} - 3 t^{3} + 3) \cdot 0 + (1 - 2 t^{3}) \cdot 0$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$t^{4} \cdot 0 - 3 t^{3} \cdot 0 + 3 \cdot 0 + (1 - 2 t^{3}) \cdot 0$

Step 4.2

Apply the distributive property.

$t^{4} \cdot 0 - 3 t^{3} \cdot 0 + 3 \cdot 0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $t^{4}$ by $0$ .

$0 - 3 t^{3} \cdot 0 + 3 \cdot 0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3.2

Multiply $0$ by $- 3$ .

$0 + 0 t^{3} + 3 \cdot 0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3.3

Multiply $0$ by $t^{3}$ .

$0 + 0 + 3 \cdot 0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3.4

Add $0$ and $0$ .

$0 + 3 \cdot 0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3.5

Multiply $3$ by $0$ .

$0 + 0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3.6

Add $0$ and $0$ .

$0 + 1 \cdot 0 - 2 t^{3} \cdot 0$

Step 4.3.7

Multiply $0$ by $1$ .

$0 + 0 - 2 t^{3} \cdot 0$

Step 4.3.8

Multiply $0$ by $- 2$ .

$0 + 0 + 0 t^{3}$

Step 4.3.9

Multiply $0$ by $t^{3}$ .

$0 + 0 + 0$

Step 4.3.10

Add $0$ and $0$ .

$0 + 0$

Step 4.3.11

Add $0$ and $0$ .

$0$