Calculus Examples

$v = (1 - t) {(1 + t^{2})}^{- 1}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} v' = \frac{d}{d t} ((1 - t) {(1 + t^{2})}^{- 1})$

Step 2

The derivative of $v'$ with respect to $t$ is $v''$ .

$v''$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $1 + t^{2}$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $1 + t^{2}$ .

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

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

By the Sum Rule, the derivative of $1 + t^{2}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [t^{2}]$ .

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

Step 3.3.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$(1 - t) (- {(1 + t^{2})}^{- 2} (0 + \frac{d}{d t} [t^{2}])) + {(1 + t^{2})}^{- 1} \frac{d}{d t} [1 - t]$

Step 3.3.3

Add $0$ and $\frac{d}{d t} [t^{2}]$ .

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

Step 3.3.4

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

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

Step 3.3.5

Multiply $2$ by $- 1$ .

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

Step 3.3.6

By the Sum Rule, the derivative of $1 - t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- t]$ .

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

Step 3.3.7

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$(1 - t) (- 2 {(1 + t^{2})}^{- 2} t) + {(1 + t^{2})}^{- 1} (0 + \frac{d}{d t} [- t])$

Step 3.3.8

Add $0$ and $\frac{d}{d t} [- t]$ .

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

Simplify the expression.

Tap for more steps...

Step 3.3.11.1

Multiply $- 1$ by $1$ .

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

Step 3.3.11.2

Move $- 1$ to the left of ${(1 + t^{2})}^{- 1}$ .

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

Step 3.3.11.3

Rewrite $- 1 {(1 + t^{2})}^{- 1}$ as $- {(1 + t^{2})}^{- 1}$ .

$(1 - t) (- 2 {(1 + t^{2})}^{- 2} t) - {(1 + t^{2})}^{- 1}$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Combine terms.

Tap for more steps...

Step 3.4.3.1

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

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

Step 3.4.3.2

Move the negative in front of the fraction.

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

Step 3.4.3.3

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

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

Step 3.4.3.4

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

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

Step 3.4.3.5

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

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

Step 3.4.3.6

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

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

Step 3.4.3.7

Combine the numerators over the common denominator.

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

Step 3.4.4

Reorder terms.

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

Step 3.4.5

Simplify the numerator.

Tap for more steps...

Step 3.4.5.1

Factor $- 1$ out of $- (1 + t^{2}) (1 - t) \frac{2 t}{{(1 + t^{2})}^{2}} - 1$ .

Tap for more steps...

Step 3.4.5.1.1

Reorder the expression.

Tap for more steps...

Step 3.4.5.1.1.1

Reorder $1$ and $t^{2}$ .

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

Step 3.4.5.1.1.2

Reorder $1$ and $- t$ .

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

Step 3.4.5.1.1.3

Reorder $1$ and $t^{2}$ .

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

Step 3.4.5.1.2

Factor $- 1$ out of $- (t^{2} + 1) (- t + 1) \frac{2 t}{{(t^{2} + 1)}^{2}}$ .

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

Step 3.4.5.1.3

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

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

Step 3.4.5.1.4

Factor $- 1$ out of $- (((t^{2} + 1) (- t + 1)) \frac{2 t}{{(t^{2} + 1)}^{2}}) - 1 (1)$ .

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

Step 3.4.5.2

Cancel the common factor of $t^{2} + 1$ .

Tap for more steps...

Step 3.4.5.2.1

Factor $t^{2} + 1$ out of ${(t^{2} + 1)}^{2}$ .

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

Step 3.4.5.2.2

Cancel the common factor.

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

Step 3.4.5.2.3

Rewrite the expression.

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

Step 3.4.5.3

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

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

Step 3.4.5.4

Move $2$ to the left of $- t + 1$ .

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

Step 3.4.5.5

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

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

Step 3.4.5.6

Combine the numerators over the common denominator.

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

Step 3.4.5.7

Reorder terms.

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

Step 3.4.5.8

Rewrite $\frac{t^{2} + 2 t (- t + 1) + 1}{t^{2} + 1}$ in a factored form.

Tap for more steps...

Step 3.4.5.8.1

Apply the distributive property.

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

Step 3.4.5.8.2

Rewrite using the commutative property of multiplication.

$\frac{- \frac{t^{2} + 2 \cdot - 1 t \cdot t + 2 t \cdot 1 + 1}{t^{2} + 1}}{t^{2} + 1}$

Step 3.4.5.8.3

Multiply $2$ by $1$ .

$\frac{- \frac{t^{2} + 2 \cdot - 1 t \cdot t + 2 t + 1}{t^{2} + 1}}{t^{2} + 1}$

Step 3.4.5.8.4

Simplify each term.

Tap for more steps...

Step 3.4.5.8.4.1

Multiply $t$ by $t$ by adding the exponents.

Tap for more steps...

Step 3.4.5.8.4.1.1

Move $t$ .

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

Step 3.4.5.8.4.1.2

Multiply $t$ by $t$ .

$\frac{- \frac{t^{2} + 2 \cdot - 1 t^{2} + 2 t + 1}{t^{2} + 1}}{t^{2} + 1}$

Step 3.4.5.8.4.2

Multiply $2$ by $- 1$ .

$\frac{- \frac{t^{2} - 2 t^{2} + 2 t + 1}{t^{2} + 1}}{t^{2} + 1}$

Step 3.4.5.8.5

Subtract $2 t^{2}$ from $t^{2}$ .

$\frac{- \frac{- t^{2} + 2 t + 1}{t^{2} + 1}}{t^{2} + 1}$

Step 3.4.6

Multiply the numerator by the reciprocal of the denominator.

$- \frac{- t^{2} + 2 t + 1}{t^{2} + 1} \cdot \frac{1}{t^{2} + 1}$

Step 3.4.7

Multiply $- \frac{- t^{2} + 2 t + 1}{t^{2} + 1} \cdot \frac{1}{t^{2} + 1}$ .

Tap for more steps...

Step 3.4.7.1

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

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

Step 3.4.7.2

Raise $t^{2} + 1$ to the power of $1$ .

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

Step 3.4.7.3

Raise $t^{2} + 1$ to the power of $1$ .

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

Step 3.4.7.4

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

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

Step 3.4.7.5

Add $1$ and $1$ .

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

Step 3.4.8

Factor $- 1$ out of $- t^{2}$ .

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

Step 3.4.9

Factor $- 1$ out of $2 t$ .

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

Step 3.4.10

Factor $- 1$ out of $- (t^{2}) - (- 2 t)$ .

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

Step 3.4.11

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

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

Step 3.4.12

Factor $- 1$ out of $- (t^{2} - 2 t) - 1 (- 1)$ .

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

Step 3.4.13

Rewrite $- (t^{2} - 2 t - 1)$ as $- 1 (t^{2} - 2 t - 1)$ .

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

Step 3.4.14

Move the negative in front of the fraction.

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

Step 3.4.15

Multiply $- 1$ by $- 1$ .

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

Step 3.4.16

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

$v'' = \frac{t^{2} - 2 t - 1}{{(t^{2} + 1)}^{2}}$

Step 5

Replace $v'$ with $\frac{d v}{d t}$ .

$\frac{d v}{d t} = \frac{t^{2} - 2 t - 1}{{(t^{2} + 1)}^{2}}$