Calculus Examples

$10.72 {(0.9 t + 10)}^{0.3}$

Step 1

Since $10.72$ is constant with respect to $t$ , the derivative of $10.72 {(0.9 t + 10)}^{0.3}$ with respect to $t$ is $10.72 \frac{d}{d t} [{(0.9 t + 10)}^{0.3}]$ .

$10.72 \frac{d}{d t} [{(0.9 t + 10)}^{0.3}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{0.3}$ and $g (t) = 0.9 t + 10$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $0.9 t + 10$ .

$10.72 (\frac{d}{d u} [u^{0.3}] \frac{d}{d t} [0.9 t + 10])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 0.3$ .

$10.72 (0.3 u^{- 0.7} \frac{d}{d t} [0.9 t + 10])$

Step 2.3

Replace all occurrences of $u$ with $0.9 t + 10$ .

$10.72 (0.3 {(0.9 t + 10)}^{- 0.7} \frac{d}{d t} [0.9 t + 10])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $0.3$ by $10.72$ .

$3.216 ({(0.9 t + 10)}^{- 0.7} \frac{d}{d t} [0.9 t + 10])$

Step 3.2

By the Sum Rule, the derivative of $0.9 t + 10$ with respect to $t$ is $\frac{d}{d t} [0.9 t] + \frac{d}{d t} [10]$ .

$3.216 {(0.9 t + 10)}^{- 0.7} (\frac{d}{d t} [0.9 t] + \frac{d}{d t} [10])$

Step 3.3

Since $0.9$ is constant with respect to $t$ , the derivative of $0.9 t$ with respect to $t$ is $0.9 \frac{d}{d t} [t]$ .

$3.216 {(0.9 t + 10)}^{- 0.7} (0.9 \frac{d}{d t} [t] + \frac{d}{d t} [10])$

Step 3.4

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

$3.216 {(0.9 t + 10)}^{- 0.7} (0.9 \cdot 1 + \frac{d}{d t} [10])$

Step 3.5

Multiply $0.9$ by $1$ .

$3.216 {(0.9 t + 10)}^{- 0.7} (0.9 + \frac{d}{d t} [10])$

Step 3.6

Since $10$ is constant with respect to $t$ , the derivative of $10$ with respect to $t$ is $0$ .

$3.216 {(0.9 t + 10)}^{- 0.7} (0.9 + 0)$

Step 3.7

Simplify the expression.

Tap for more steps...

Step 3.7.1

Add $0.9$ and $0$ .

$3.216 {(0.9 t + 10)}^{- 0.7} \cdot 0.9$

Step 3.7.2

Multiply $0.9$ by $3.216$ .

$2.8944 {(0.9 t + 10)}^{- 0.7}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2.8944 \frac{1}{{(0.9 t + 10)}^{0.7}}$

Step 5

Combine $2.8944$ and $\frac{1}{{(0.9 t + 10)}^{0.7}}$ .

$\frac{2.8944}{{(0.9 t + 10)}^{0.7}}$