Calculus Examples

$s = 5 t^{4} - 4 t^{5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (5 t^{4} - 4 t^{5})$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{d}{d t} [5 t^{4}] + \frac{d}{d t} [- 4 t^{5}]$

Step 3.2

Evaluate $\frac{d}{d t} [5 t^{4}]$ .

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Step 3.2.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{4}$ with respect to $t$ is $5 \frac{d}{d t} [t^{4}]$ .

$5 \frac{d}{d t} [t^{4}] + \frac{d}{d t} [- 4 t^{5}]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

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

Step 3.2.3

Multiply $4$ by $5$ .

$20 t^{3} + \frac{d}{d t} [- 4 t^{5}]$

Step 3.3

Evaluate $\frac{d}{d t} [- 4 t^{5}]$ .

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Step 3.3.1

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4 t^{5}$ with respect to $t$ is $- 4 \frac{d}{d t} [t^{5}]$ .

$20 t^{3} - 4 \frac{d}{d t} [t^{5}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$20 t^{3} - 4 (5 t^{4})$

Step 3.3.3

Multiply $5$ by $- 4$ .

$20 t^{3} - 20 t^{4}$

Step 3.4

Reorder terms.

$- 20 t^{4} + 20 t^{3}$

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = - 20 t^{4} + 20 t^{3}$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = - 20 t^{4} + 20 t^{3}$