Calculus Examples

$y = \frac{1}{3} \cdot (t^{6} - t^{3})$

Step 1

Since $\frac{1}{3}$ is constant with respect to $t$ , the derivative of $\frac{1}{3} (t^{6} - t^{3})$ with respect to $t$ is $\frac{1}{3} \frac{d}{d t} [t^{6} - t^{3}]$ .

$\frac{1}{3} \frac{d}{d t} [t^{6} - t^{3}]$

Step 2

By the Sum Rule, the derivative of $t^{6} - t^{3}$ with respect to $t$ is $\frac{d}{d t} [t^{6}] + \frac{d}{d t} [- t^{3}]$ .

$\frac{1}{3} (\frac{d}{d t} [t^{6}] + \frac{d}{d t} [- t^{3}])$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 6$ .

$\frac{1}{3} (6 t^{5} + \frac{d}{d t} [- t^{3}])$

Step 4

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{3}$ with respect to $t$ is $- \frac{d}{d t} [t^{3}]$ .

$\frac{1}{3} (6 t^{5} - \frac{d}{d t} [t^{3}])$

Step 5

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

$\frac{1}{3} (6 t^{5} - (3 t^{2}))$

Step 6

Multiply $3$ by $- 1$ .

$\frac{1}{3} (6 t^{5} - 3 t^{2})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{3} (6 t^{5}) + \frac{1}{3} (- 3 t^{2})$

Step 7.2

Combine terms.

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Step 7.2.1

Combine $6$ and $\frac{1}{3}$ .

$\frac{6}{3} t^{5} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.2

Combine $\frac{6}{3}$ and $t^{5}$ .

$\frac{6 t^{5}}{3} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.3

Cancel the common factor of $6$ and $3$ .

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Step 7.2.3.1

Factor $3$ out of $6 t^{5}$ .

$\frac{3 (2 t^{5})}{3} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 t^{5})}{3 (1)} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.3.2.2

Cancel the common factor.

$\frac{3 (2 t^{5})}{3 \cdot 1} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.3.2.3

Rewrite the expression.

$\frac{2 t^{5}}{1} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.3.2.4

Divide $2 t^{5}$ by $1$ .

$2 t^{5} + \frac{1}{3} (- 3 t^{2})$

Step 7.2.4

Combine $- 3$ and $\frac{1}{3}$ .

$2 t^{5} + \frac{- 3}{3} t^{2}$

Step 7.2.5

Combine $\frac{- 3}{3}$ and $t^{2}$ .

$2 t^{5} + \frac{- 3 t^{2}}{3}$

Step 7.2.6

Cancel the common factor of $- 3$ and $3$ .

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Step 7.2.6.1

Factor $3$ out of $- 3 t^{2}$ .

$2 t^{5} + \frac{3 (- t^{2})}{3}$

Step 7.2.6.2

Cancel the common factors.

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Step 7.2.6.2.1

Factor $3$ out of $3$ .

$2 t^{5} + \frac{3 (- t^{2})}{3 (1)}$

Step 7.2.6.2.2

Cancel the common factor.

$2 t^{5} + \frac{3 (- t^{2})}{3 \cdot 1}$

Step 7.2.6.2.3

Rewrite the expression.

$2 t^{5} + \frac{- t^{2}}{1}$

Step 7.2.6.2.4

Divide $- t^{2}$ by $1$ .

$2 t^{5} - t^{2}$