Calculus Examples

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

Step 1

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 $g (t) = {(t - 3)}^{2}$ .

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

Multiply the exponents in ${({(t - 3)}^{2})}^{2}$ .

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Step 2.1.1

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

$\frac{{(t - 3)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 3)}^{2}]}{{(t - 3)}^{2 \cdot 2}}$

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{{(t - 3)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 3)}^{2}]}{{(t - 3)}^{4}}$

Step 2.2

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

$\frac{{(t - 3)}^{2} \cdot 1 - t \frac{d}{d t} [{(t - 3)}^{2}]}{{(t - 3)}^{4}}$

Step 2.3

Multiply ${(t - 3)}^{2}$ by $1$ .

$\frac{{(t - 3)}^{2} - t \frac{d}{d t} [{(t - 3)}^{2}]}{{(t - 3)}^{4}}$

Step 3

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 $g (t) = t - 3$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $t - 3$ .

$\frac{{(t - 3)}^{2} - t (\frac{d}{d u} [u^{2}] \frac{d}{d t} [t - 3])}{{(t - 3)}^{4}}$

Step 3.2

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

$\frac{{(t - 3)}^{2} - t (2 u \frac{d}{d t} [t - 3])}{{(t - 3)}^{4}}$

Step 3.3

Replace all occurrences of $u$ with $t - 3$ .

$\frac{{(t - 3)}^{2} - t (2 (t - 3) \frac{d}{d t} [t - 3])}{{(t - 3)}^{4}}$

Step 4

Simplify with factoring out.

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Step 4.1

Multiply $2$ by $- 1$ .

$\frac{{(t - 3)}^{2} - 2 t ((t - 3) \frac{d}{d t} [t - 3])}{{(t - 3)}^{4}}$

Step 4.2

Factor $t - 3$ out of ${(t - 3)}^{2} - 2 t ((t - 3) \frac{d}{d t} [t - 3])$ .

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Step 4.2.1

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

$\frac{(t - 3) (t - 3) - 2 t ((t - 3) \frac{d}{d t} [t - 3])}{{(t - 3)}^{4}}$

Step 4.2.2

Factor $t - 3$ out of $- 2 t ((t - 3) \frac{d}{d t} [t - 3])$ .

$\frac{(t - 3) (t - 3) + (t - 3) (- 2 t (\frac{d}{d t} [t - 3]))}{{(t - 3)}^{4}}$

Step 4.2.3

Factor $t - 3$ out of $(t - 3) (t - 3) + (t - 3) (- 2 t (\frac{d}{d t} [t - 3]))$ .

$\frac{(t - 3) (t - 3 - 2 t (\frac{d}{d t} [t - 3]))}{{(t - 3)}^{4}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $t - 3$ out of ${(t - 3)}^{4}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

By the Sum Rule, the derivative of $t - 3$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 3]$ .

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

Step 7

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

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

Step 8

Since $- 3$ is constant with respect to $t$ , the derivative of $- 3$ with respect to $t$ is $0$ .

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

Step 9

Simplify by adding terms.

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Step 9.1

Add $1$ and $0$ .

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

Step 9.2

Multiply $- 2$ by $1$ .

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

Step 9.3

Subtract $2 t$ from $t$ .

$\frac{- t - 3}{{(t - 3)}^{3}}$

Step 10

Simplify.

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Step 10.1

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

$\frac{- (t) - 3}{{(t - 3)}^{3}}$

Step 10.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 10.3

Factor $- 1$ out of $- (t) - 1 (3)$ .

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

Step 10.4

Rewrite $- (t + 3)$ as $- 1 (t + 3)$ .

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

Step 10.5

Move the negative in front of the fraction.

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