Calculus Examples

$g (t) = \frac{7}{8 t^{6}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Since $\frac{7}{8}$ is constant with respect to $t$ , the derivative of $\frac{7}{8 t^{6}}$ with respect to $t$ is $\frac{7}{8} \frac{d}{d t} [\frac{1}{t^{6}}]$ .

$\frac{7}{8} \frac{d}{d t} [\frac{1}{t^{6}}]$

Step 1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1

Rewrite $\frac{1}{t^{6}}$ as ${(t^{6})}^{- 1}$ .

$\frac{7}{8} \frac{d}{d t} [{(t^{6})}^{- 1}]$

Step 1.2.2

Multiply the exponents in ${(t^{6})}^{- 1}$ .

Tap for more steps...

Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{7}{8} \frac{d}{d t} [t^{6 \cdot - 1}]$

Step 1.2.2.2

Multiply $6$ by $- 1$ .

$\frac{7}{8} \frac{d}{d t} [t^{- 6}]$

Step 1.3

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

$\frac{7}{8} (- 6 t^{- 7})$

Step 1.4

Simplify terms.

Tap for more steps...

Step 1.4.1

Combine $- 6$ and $\frac{7}{8}$ .

$\frac{- 6 \cdot 7}{8} t^{- 7}$

Step 1.4.2

Multiply $- 6$ by $7$ .

$\frac{- 42}{8} t^{- 7}$

Step 1.4.3

Combine $\frac{- 42}{8}$ and $t^{- 7}$ .

$\frac{- 42 t^{- 7}}{8}$

Step 1.4.4

Move $t^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 42}{8 t^{7}}$

Step 1.4.5

Cancel the common factor of $- 42$ and $8$ .

Tap for more steps...

Step 1.4.5.1

Factor $2$ out of $- 42$ .

$\frac{2 (- 21)}{8 t^{7}}$

Step 1.4.5.2

Cancel the common factors.

Tap for more steps...

Step 1.4.5.2.1

Factor $2$ out of $8 t^{7}$ .

$\frac{2 (- 21)}{2 (4 t^{7})}$

Step 1.4.5.2.2

Cancel the common factor.

$\frac{2 \cdot - 21}{2 (4 t^{7})}$

Step 1.4.5.2.3

Rewrite the expression.

$\frac{- 21}{4 t^{7}}$

Step 1.4.6

Move the negative in front of the fraction.

$f' (t) = - \frac{21}{4 t^{7}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $- \frac{21}{4}$ is constant with respect to $t$ , the derivative of $- \frac{21}{4 t^{7}}$ with respect to $t$ is $- \frac{21}{4} \frac{d}{d t} [\frac{1}{t^{7}}]$ .

$- \frac{21}{4} \frac{d}{d t} [\frac{1}{t^{7}}]$

Step 2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1

Rewrite $\frac{1}{t^{7}}$ as ${(t^{7})}^{- 1}$ .

$- \frac{21}{4} \frac{d}{d t} [{(t^{7})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(t^{7})}^{- 1}$ .

Tap for more steps...

Step 2.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{21}{4} \frac{d}{d t} [t^{7 \cdot - 1}]$

Step 2.2.2.2

Multiply $7$ by $- 1$ .

$- \frac{21}{4} \frac{d}{d t} [t^{- 7}]$

Step 2.3

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

$- \frac{21}{4} (- 7 t^{- 8})$

Step 2.4

Combine fractions.

Tap for more steps...

Step 2.4.1

Multiply $- 7$ by $- 1$ .

$7 (\frac{21}{4}) t^{- 8}$

Step 2.4.2

Combine $7$ and $\frac{21}{4}$ .

$\frac{7 \cdot 21}{4} t^{- 8}$

Step 2.4.3

Multiply $7$ by $21$ .

$\frac{147}{4} t^{- 8}$

Step 2.4.4

Combine $\frac{147}{4}$ and $t^{- 8}$ .

$\frac{147 t^{- 8}}{4}$

Step 2.4.5

Move $t^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (t) = \frac{147}{4 t^{8}}$

Step 3

The second derivative of $g (t)$ with respect to $t$ is $\frac{147}{4 t^{8}}$ .

$\frac{147}{4 t^{8}}$