Calculus Examples

$y = \frac{100 t}{{(t + 20)}^{2}}$

Step 1

Since $100$ is constant with respect to $t$ , the derivative of $\frac{100 t}{{(t + 20)}^{2}}$ with respect to $t$ is $100 \frac{d}{d t} [\frac{t}{{(t + 20)}^{2}}]$ .

$100 \frac{d}{d t} [\frac{t}{{(t + 20)}^{2}}]$

Step 2

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 $g (t) = {(t + 20)}^{2}$ .

$100 \frac{{(t + 20)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 20)}^{2}]}{{({(t + 20)}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

Multiply the exponents in ${({(t + 20)}^{2})}^{2}$ .

Tap for more steps...

Step 3.1.1

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

$100 \frac{{(t + 20)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 20)}^{2}]}{{(t + 20)}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$100 \frac{{(t + 20)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t + 20)}^{2}]}{{(t + 20)}^{4}}$

Step 3.2

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

$100 \frac{{(t + 20)}^{2} \cdot 1 - t \frac{d}{d t} [{(t + 20)}^{2}]}{{(t + 20)}^{4}}$

Step 3.3

Multiply ${(t + 20)}^{2}$ by $1$ .

$100 \frac{{(t + 20)}^{2} - t \frac{d}{d t} [{(t + 20)}^{2}]}{{(t + 20)}^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = t + 20$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u$ as $t + 20$ .

$100 \frac{{(t + 20)}^{2} - t (\frac{d}{d u} [u^{2}] \frac{d}{d t} [t + 20])}{{(t + 20)}^{4}}$

Step 4.2

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

$100 \frac{{(t + 20)}^{2} - t (2 u \frac{d}{d t} [t + 20])}{{(t + 20)}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $t + 20$ .

$100 \frac{{(t + 20)}^{2} - t (2 (t + 20) \frac{d}{d t} [t + 20])}{{(t + 20)}^{4}}$

Step 5

Simplify with factoring out.

Tap for more steps...

Step 5.1

Multiply $2$ by $- 1$ .

$100 \frac{{(t + 20)}^{2} - 2 t ((t + 20) \frac{d}{d t} [t + 20])}{{(t + 20)}^{4}}$

Step 5.2

Factor $t + 20$ out of ${(t + 20)}^{2} - 2 t ((t + 20) \frac{d}{d t} [t + 20])$ .

Tap for more steps...

Step 5.2.1

Factor $t + 20$ out of ${(t + 20)}^{2}$ .

$100 \frac{(t + 20) (t + 20) - 2 t ((t + 20) \frac{d}{d t} [t + 20])}{{(t + 20)}^{4}}$

Step 5.2.2

Factor $t + 20$ out of $- 2 t ((t + 20) \frac{d}{d t} [t + 20])$ .

$100 \frac{(t + 20) (t + 20) + (t + 20) (- 2 t (\frac{d}{d t} [t + 20]))}{{(t + 20)}^{4}}$

Step 5.2.3

Factor $t + 20$ out of $(t + 20) (t + 20) + (t + 20) (- 2 t (\frac{d}{d t} [t + 20]))$ .

$100 \frac{(t + 20) (t + 20 - 2 t (\frac{d}{d t} [t + 20]))}{{(t + 20)}^{4}}$

Step 6

Cancel the common factors.

Tap for more steps...

Step 6.1

Factor $t + 20$ out of ${(t + 20)}^{4}$ .

$100 \frac{(t + 20) (t + 20 - 2 t (\frac{d}{d t} [t + 20]))}{(t + 20) {(t + 20)}^{3}}$

Step 6.2

Cancel the common factor.

$100 \frac{(t + 20) (t + 20 - 2 t (\frac{d}{d t} [t + 20]))}{(t + 20) {(t + 20)}^{3}}$

Step 6.3

Rewrite the expression.

$100 \frac{t + 20 - 2 t (\frac{d}{d t} [t + 20])}{{(t + 20)}^{3}}$

Step 7

By the Sum Rule, the derivative of $t + 20$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [20]$ .

$100 \frac{t + 20 - 2 t (\frac{d}{d t} [t] + \frac{d}{d t} [20])}{{(t + 20)}^{3}}$

Step 8

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

$100 \frac{t + 20 - 2 t (1 + \frac{d}{d t} [20])}{{(t + 20)}^{3}}$

Step 9

Since $20$ is constant with respect to $t$ , the derivative of $20$ with respect to $t$ is $0$ .

$100 \frac{t + 20 - 2 t (1 + 0)}{{(t + 20)}^{3}}$

Step 10

Simplify terms.

Tap for more steps...

Step 10.1

Add $1$ and $0$ .

$100 \frac{t + 20 - 2 t \cdot 1}{{(t + 20)}^{3}}$

Step 10.2

Multiply $- 2$ by $1$ .

$100 \frac{t + 20 - 2 t}{{(t + 20)}^{3}}$

Step 10.3

Subtract $2 t$ from $t$ .

$100 \frac{- t + 20}{{(t + 20)}^{3}}$

Step 10.4

Combine $100$ and $\frac{- t + 20}{{(t + 20)}^{3}}$ .

$\frac{100 (- t + 20)}{{(t + 20)}^{3}}$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

$\frac{100 (- t) + 100 \cdot 20}{{(t + 20)}^{3}}$

Step 11.2

Simplify each term.

Tap for more steps...

Step 11.2.1

Multiply $- 1$ by $100$ .

$\frac{- 100 t + 100 \cdot 20}{{(t + 20)}^{3}}$

Step 11.2.2

Multiply $100$ by $20$ .

$\frac{- 100 t + 2000}{{(t + 20)}^{3}}$

Step 11.3

Factor $100$ out of $- 100 t + 2000$ .

Tap for more steps...

Step 11.3.1

Factor $100$ out of $- 100 t$ .

$\frac{100 (- t) + 2000}{{(t + 20)}^{3}}$

Step 11.3.2

Factor $100$ out of $2000$ .

$\frac{100 (- t) + 100 (20)}{{(t + 20)}^{3}}$

Step 11.3.3

Factor $100$ out of $100 (- t) + 100 (20)$ .

$\frac{100 (- t + 20)}{{(t + 20)}^{3}}$

Step 11.4

Factor $- 1$ out of $- t$ .

$\frac{100 (- (t) + 20)}{{(t + 20)}^{3}}$

Step 11.5

Rewrite $20$ as $- 1 (- 20)$ .

$\frac{100 (- (t) - 1 (- 20))}{{(t + 20)}^{3}}$

Step 11.6

Factor $- 1$ out of $- (t) - 1 (- 20)$ .

$\frac{100 (- (t - 20))}{{(t + 20)}^{3}}$

Step 11.7

Rewrite $- (t - 20)$ as $- 1 (t - 20)$ .

$\frac{100 (- 1 (t - 20))}{{(t + 20)}^{3}}$

Step 11.8

Move the negative in front of the fraction.

$- \frac{100 (t - 20)}{{(t + 20)}^{3}}$