Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the Constant Rule.

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Step 2.1

Since $2 t$ is constant with respect to $v$ , the derivative of $2 t$ with respect to $v$ is $0$ .

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

Step 2.2

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

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

Step 3

Expand $(t + 1) (t + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $t$ by $t$ .

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

Step 4.1.2

Multiply $t$ by $1$ .

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

Step 4.1.3

Multiply $t$ by $1$ .

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

Step 4.1.4

Multiply $1$ by $1$ .

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

Step 4.2

Add $t$ and $t$ .

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

Step 5

Since $t^{2} + 2 t + 1$ is constant with respect to $v$ , the derivative of $t^{2} + 2 t + 1$ with respect to $v$ is $0$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Simplify each term.

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Step 6.1.1.1

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

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

Step 6.1.1.2

Expand $(t + 1) (t + 1)$ using the FOIL Method.

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Step 6.1.1.2.1

Apply the distributive property.

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

Step 6.1.1.2.2

Apply the distributive property.

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

Step 6.1.1.2.3

Apply the distributive property.

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

Step 6.1.1.3

Simplify and combine like terms.

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Step 6.1.1.3.1

Simplify each term.

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Step 6.1.1.3.1.1

Multiply $t$ by $t$ .

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

Step 6.1.1.3.1.2

Multiply $t$ by $1$ .

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

Step 6.1.1.3.1.3

Multiply $t$ by $1$ .

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

Step 6.1.1.3.1.4

Multiply $1$ by $1$ .

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

Step 6.1.1.3.2

Add $t$ and $t$ .

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

Step 6.1.1.4

Multiply $t^{2} + 2 t + 1$ by $0$ .

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

Step 6.1.1.5

Multiply $- 1$ by $2$ .

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

Step 6.1.1.6

Multiply $- 2 t \cdot 0$ .

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Step 6.1.1.6.1

Multiply $0$ by $- 2$ .

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

Step 6.1.1.6.2

Multiply $0$ by $t$ .

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

Step 6.1.2

Add $0$ and $0$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $2$ .

$\frac{0}{{(t + 1)}^{4}}$

Step 6.3

Divide $0$ by ${(t + 1)}^{4}$ .

$0$