Calculus Examples

$s = \frac{t^{2} + 5 t - 1}{t^{2}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d s} [\frac{f (s)}{g (s)}]$ is $\frac{g (s) \frac{d}{d s} [f (s)] - f (s) \frac{d}{d s} [g (s)]}{{g (s)}^{2}}$ where $f (s) = t^{2} + 5 t - 1$ and $g (s) = t^{2}$ .

$\frac{t^{2} \frac{d}{d s} [t^{2} + 5 t - 1] - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $t^{2} + 5 t - 1$ with respect to $s$ is $\frac{d}{d s} [t^{2}] + \frac{d}{d s} [5 t] + \frac{d}{d s} [- 1]$ .

$\frac{t^{2} (\frac{d}{d s} [t^{2}] + \frac{d}{d s} [5 t] + \frac{d}{d s} [- 1]) - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 2.2

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

$\frac{t^{2} (0 + \frac{d}{d s} [5 t] + \frac{d}{d s} [- 1]) - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 2.3

Since $5 t$ is constant with respect to $s$ , the derivative of $5 t$ with respect to $s$ is $0$ .

$\frac{t^{2} (0 + 0 + \frac{d}{d s} [- 1]) - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 2.4

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

$\frac{t^{2} (0 + 0 + 0) - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 3

Combine terms.

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Step 3.1

Add $0$ and $0$ .

$\frac{t^{2} (0 + 0) - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 3.2

Add $0$ and $0$ .

$\frac{t^{2} \cdot 0 - (t^{2} + 5 t - 1) \frac{d}{d s} [t^{2}]}{{(t^{2})}^{2}}$

Step 4

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

$\frac{t^{2} \cdot 0 - (t^{2} + 5 t - 1) \cdot 0}{{(t^{2})}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{t^{2} \cdot 0 + (- t^{2} - (5 t) - - 1) \cdot 0}{{(t^{2})}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{t^{2} \cdot 0 - t^{2} \cdot 0 - (5 t) \cdot 0 - - 1 \cdot 0}{{(t^{2})}^{2}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Subtract $t^{2} \cdot 0$ from $t^{2} \cdot 0$ .

$\frac{0 - (5 t) \cdot 0 - - 1 \cdot 0}{{(t^{2})}^{2}}$

Step 5.3.2

Subtract $5 t \cdot 0$ from $0$ .

$\frac{- (5 t) \cdot 0 - - 1 \cdot 0}{{(t^{2})}^{2}}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Multiply $5$ by $- 1$ .

$\frac{- 5 t \cdot 0 - - 1 \cdot 0}{{(t^{2})}^{2}}$

Step 5.3.3.2

Multiply $- 5 t \cdot 0$ .

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Step 5.3.3.2.1

Multiply $0$ by $- 5$ .

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

Step 5.3.3.2.2

Multiply $0$ by $t$ .

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

Step 5.3.3.3

Multiply $- - 1 \cdot 0$ .

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Step 5.3.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.3.2

Multiply $0$ by $1$ .

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

Step 5.3.4

Add $0$ and $0$ .

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

Step 5.4

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

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Step 5.4.1

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

$\frac{0}{t^{2 \cdot 2}}$

Step 5.4.2

Multiply $2$ by $2$ .

$\frac{0}{t^{4}}$

Step 5.5

Divide $0$ by $t^{4}$ .

$0$