Calculus Examples

$q = \sin (\frac{t}{\sqrt{t + 9}})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \sin (t)$ and $g (t) = \frac{t}{\sqrt{t + 9}}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{t}{\sqrt{t + 9}}$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d t} [\frac{t}{\sqrt{t + 9}}]$

Step 1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d t} [\frac{t}{\sqrt{t + 9}}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{t}{\sqrt{t + 9}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{d}{d t} [\frac{t}{\sqrt{t + 9}}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 9}$ as ${(t + 9)}^{\frac{1}{2}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{d}{d t} [\frac{t}{{(t + 9)}^{\frac{1}{2}}}]$

Step 3

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t$ and $g (t) = {(t + 9)}^{\frac{1}{2}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{{({(t + 9)}^{\frac{1}{2}})}^{2}}$

Step 4

Multiply the exponents in ${({(t + 9)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.1

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

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{{(t + 9)}^{\frac{1}{2} \cdot 2}}$

Step 4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1

Cancel the common factor.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{{(t + 9)}^{\frac{1}{2} \cdot 2}}$

Step 4.2.2

Rewrite the expression.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{{(t + 9)}^{1}}$

Step 5

Simplify.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{t + 9}$

Step 6

Differentiate using the Power Rule.

Tap for more steps...

Step 6.1

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

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \cdot 1 - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{t + 9}$

Step 6.2

Multiply ${(t + 9)}^{\frac{1}{2}}$ by $1$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t \frac{d}{d t} [{(t + 9)}^{\frac{1}{2}}]}{t + 9}$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = t + 9$ .

Tap for more steps...

Step 7.1

To apply the Chain Rule, set $u_{2}$ as $t + 9$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d t} [t + 9])}{t + 9}$

Step 7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 9])}{t + 9}$

Step 7.3

Replace all occurrences of $u_{2}$ with $t + 9$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 9])}{t + 9}$

Step 8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 9

Combine $- 1$ and $\frac{2}{2}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 10

Combine the numerators over the common denominator.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 11

Simplify the numerator.

Tap for more steps...

Step 11.1

Multiply $- 1$ by $2$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{\frac{1 - 2}{2}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 11.2

Subtract $2$ from $1$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{\frac{- 1}{2}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 12

Combine fractions.

Tap for more steps...

Step 12.1

Move the negative in front of the fraction.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2} {(t + 9)}^{- \frac{1}{2}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 12.2

Combine $\frac{1}{2}$ and ${(t + 9)}^{- \frac{1}{2}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{{(t + 9)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [t + 9])}{t + 9}$

Step 12.3

Move ${(t + 9)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - t (\frac{1}{2 {(t + 9)}^{\frac{1}{2}}} \frac{d}{d t} [t + 9])}{t + 9}$

Step 12.4

Combine $\frac{1}{2 {(t + 9)}^{\frac{1}{2}}}$ and $t$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}} \frac{d}{d t} [t + 9]}{t + 9}$

Step 13

By the Sum Rule, the derivative of $t + 9$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [9]$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}} (\frac{d}{d t} [t] + \frac{d}{d t} [9])}{t + 9}$

Step 14

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

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}} (1 + \frac{d}{d t} [9])}{t + 9}$

Step 15

Since $9$ is constant with respect to $t$ , the derivative of $9$ with respect to $t$ is $0$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}} (1 + 0)}{t + 9}$

Step 16

Simplify the expression.

Tap for more steps...

Step 16.1

Add $1$ and $0$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}} \cdot 1}{t + 9}$

Step 16.2

Multiply $- 1$ by $1$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 17

To write ${(t + 9)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(t + 9)}^{\frac{1}{2}}}{2 {(t + 9)}^{\frac{1}{2}}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{{(t + 9)}^{\frac{1}{2}} \cdot \frac{2 {(t + 9)}^{\frac{1}{2}}}{2 {(t + 9)}^{\frac{1}{2}}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 18

Combine ${(t + 9)}^{\frac{1}{2}}$ and $\frac{2 {(t + 9)}^{\frac{1}{2}}}{2 {(t + 9)}^{\frac{1}{2}}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{\frac{1}{2}} (2 {(t + 9)}^{\frac{1}{2}})}{2 {(t + 9)}^{\frac{1}{2}}} - \frac{t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 19

Combine the numerators over the common denominator.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{\frac{1}{2}} (2 {(t + 9)}^{\frac{1}{2}}) - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 20

Multiply ${(t + 9)}^{\frac{1}{2}}$ by ${(t + 9)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 20.1

Move ${(t + 9)}^{\frac{1}{2}}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{\frac{1}{2}} {(t + 9)}^{\frac{1}{2}} \cdot 2 - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 20.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2 - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 20.3

Combine the numerators over the common denominator.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{\frac{1 + 1}{2}} \cdot 2 - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 20.4

Add $1$ and $1$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{\frac{2}{2}} \cdot 2 - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 20.5

Divide $2$ by $2$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{{(t + 9)}^{1} \cdot 2 - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 21

Simplify ${(t + 9)}^{1} \cdot 2$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{(t + 9) \cdot 2 - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 22

Move $2$ to the left of $t + 9$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{\frac{2 \cdot (t + 9) - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$

Step 23

Rewrite $\frac{\frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{1}{2}}}}{t + 9}$ as a product.

$\cos (\frac{t}{\sqrt{t + 9}}) (\frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{1}{2}}} \cdot \frac{1}{t + 9})$

Step 24

Multiply $\frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{1}{2}}}$ by $\frac{1}{t + 9}$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{1}{2}} (t + 9)}$

Step 25

Raise $t + 9$ to the power of $1$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 (t + 9) - t}{2 ({(t + 9)}^{1} {(t + 9)}^{\frac{1}{2}})}$

Step 26

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 (t + 9) - t}{2 {(t + 9)}^{1 + \frac{1}{2}}}$

Step 27

Write $1$ as a fraction with a common denominator.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 28

Combine the numerators over the common denominator.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{2 + 1}{2}}}$

Step 29

Add $2$ and $1$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 (t + 9) - t}{2 {(t + 9)}^{\frac{3}{2}}}$

Step 30

Simplify.

Tap for more steps...

Step 30.1

Apply the distributive property.

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 t + 2 \cdot 9 - t}{2 {(t + 9)}^{\frac{3}{2}}}$

Step 30.2

Simplify the numerator.

Tap for more steps...

Step 30.2.1

Multiply $2$ by $9$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{2 t + 18 - t}{2 {(t + 9)}^{\frac{3}{2}}}$

Step 30.2.2

Subtract $t$ from $2 t$ .

$\cos (\frac{t}{\sqrt{t + 9}}) \frac{t + 18}{2 {(t + 9)}^{\frac{3}{2}}}$

Step 31

Combine $\cos (\frac{t}{\sqrt{t + 9}})$ and $\frac{t + 18}{2 {(t + 9)}^{\frac{3}{2}}}$ .

$\frac{\cos (\frac{t}{\sqrt{t + 9}}) (t + 18)}{2 {(t + 9)}^{\frac{3}{2}}}$