Calculus Examples

$h (t) = \frac{t^{9} - 7 t^{3} + 6 t^{2}}{6 t^{2}}$

Step 1

Since $\frac{1}{6}$ is constant with respect to $t$ , the derivative of $\frac{t^{9} - 7 t^{3} + 6 t^{2}}{6 t^{2}}$ with respect to $t$ is $\frac{1}{6} \frac{d}{d t} [\frac{t^{9} - 7 t^{3} + 6 t^{2}}{t^{2}}]$ .

$\frac{1}{6} \frac{d}{d t} [\frac{t^{9} - 7 t^{3} + 6 t^{2}}{t^{2}}]$

Step 2

Simplify terms.

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Step 2.1

Factor $t^{2}$ out of $t^{9} - 7 t^{3} + 6 t^{2}$ .

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Step 2.1.1

Factor $t^{2}$ out of $t^{9}$ .

$\frac{1}{6} \frac{d}{d t} [\frac{t^{2} t^{7} - 7 t^{3} + 6 t^{2}}{t^{2}}]$

Step 2.1.2

Factor $t^{2}$ out of $- 7 t^{3}$ .

$\frac{1}{6} \frac{d}{d t} [\frac{t^{2} t^{7} + t^{2} (- 7 t) + 6 t^{2}}{t^{2}}]$

Step 2.1.3

Factor $t^{2}$ out of $6 t^{2}$ .

$\frac{1}{6} \frac{d}{d t} [\frac{t^{2} t^{7} + t^{2} (- 7 t) + t^{2} \cdot 6}{t^{2}}]$

Step 2.1.4

Factor $t^{2}$ out of $t^{2} t^{7} + t^{2} (- 7 t)$ .

$\frac{1}{6} \frac{d}{d t} [\frac{t^{2} (t^{7} - 7 t) + t^{2} \cdot 6}{t^{2}}]$

Step 2.1.5

Factor $t^{2}$ out of $t^{2} (t^{7} - 7 t) + t^{2} \cdot 6$ .

$\frac{1}{6} \frac{d}{d t} [\frac{t^{2} (t^{7} - 7 t + 6)}{t^{2}}]$

Step 2.2

Cancel the common factor of $t^{2}$ .

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Step 2.2.1

Cancel the common factor.

$\frac{1}{6} \frac{d}{d t} [\frac{t^{2} (t^{7} - 7 t + 6)}{t^{2}}]$

Step 2.2.2

Divide $t^{7} - 7 t + 6$ by $1$ .

$\frac{1}{6} \frac{d}{d t} [t^{7} - 7 t + 6]$

Step 3

By the Sum Rule, the derivative of $t^{7} - 7 t + 6$ with respect to $t$ is $\frac{d}{d t} [t^{7}] + \frac{d}{d t} [- 7 t] + \frac{d}{d t} [6]$ .

$\frac{1}{6} (\frac{d}{d t} [t^{7}] + \frac{d}{d t} [- 7 t] + \frac{d}{d t} [6])$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$\frac{1}{6} (7 t^{6} + \frac{d}{d t} [- 7 t] + \frac{d}{d t} [6])$

Step 5

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 t$ with respect to $t$ is $- 7 \frac{d}{d t} [t]$ .

$\frac{1}{6} (7 t^{6} - 7 \frac{d}{d t} [t] + \frac{d}{d t} [6])$

Step 6

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

$\frac{1}{6} (7 t^{6} - 7 \cdot 1 + \frac{d}{d t} [6])$

Step 7

Multiply $- 7$ by $1$ .

$\frac{1}{6} (7 t^{6} - 7 + \frac{d}{d t} [6])$

Step 8

Since $6$ is constant with respect to $t$ , the derivative of $6$ with respect to $t$ is $0$ .

$\frac{1}{6} (7 t^{6} - 7 + 0)$

Step 9

Add $7 t^{6} - 7$ and $0$ .

$\frac{1}{6} (7 t^{6} - 7)$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{1}{6} (7 t^{6}) + \frac{1}{6} \cdot - 7$

Step 10.2

Combine terms.

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Step 10.2.1

Combine $7$ and $\frac{1}{6}$ .

$\frac{7}{6} t^{6} + \frac{1}{6} \cdot - 7$

Step 10.2.2

Combine $\frac{7}{6}$ and $t^{6}$ .

$\frac{7 t^{6}}{6} + \frac{1}{6} \cdot - 7$

Step 10.2.3

Combine $\frac{1}{6}$ and $- 7$ .

$\frac{7 t^{6}}{6} + \frac{- 7}{6}$

Step 10.2.4

Move the negative in front of the fraction.

$\frac{7 t^{6}}{6} - \frac{7}{6}$