Calculus Examples

$s = t^{4} \tan (t) - \sqrt{t}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$s = t^{4} \tan (t) - t^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (t^{4} \tan (t) - t^{\frac{1}{2}})$

Step 3

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

$s'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $t^{4} \tan (t) - t^{\frac{1}{2}}$ with respect to $t$ is $\frac{d}{d t} [t^{4} \tan (t)] + \frac{d}{d t} [- t^{\frac{1}{2}}]$ .

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

Step 4.2

Evaluate $\frac{d}{d t} [t^{4} \tan (t)]$ .

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Step 4.2.1

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 $f (t) = t^{4}$ and $g (t) = \tan (t)$ .

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

Step 4.2.2

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

$t^{4} \sec^{2} (t) + \tan (t) \frac{d}{d t} [t^{4}] + \frac{d}{d t} [- t^{\frac{1}{2}}]$

Step 4.2.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

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

Step 4.3

Evaluate $\frac{d}{d t} [- t^{\frac{1}{2}}]$ .

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Step 4.3.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{\frac{1}{2}}$ with respect to $t$ is $- \frac{d}{d t} [t^{\frac{1}{2}}]$ .

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

Step 4.3.2

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

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{\frac{1}{2} - 1})$

Step 4.3.3

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

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 4.3.4

Combine $- 1$ and $\frac{2}{2}$ .

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.3.5

Combine the numerators over the common denominator.

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}})$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Multiply $- 1$ by $2$ .

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{\frac{1 - 2}{2}})$

Step 4.3.6.2

Subtract $2$ from $1$ .

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{\frac{- 1}{2}})$

Step 4.3.7

Move the negative in front of the fraction.

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - (\frac{1}{2} t^{- \frac{1}{2}})$

Step 4.3.8

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

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - \frac{t^{- \frac{1}{2}}}{2}$

Step 4.3.9

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$t^{4} \sec^{2} (t) + \tan (t) (4 t^{3}) - \frac{1}{2 t^{\frac{1}{2}}}$

Step 4.4

Reorder terms.

$t^{4} \sec^{2} (t) + 4 t^{3} \tan (t) - \frac{1}{2 t^{\frac{1}{2}}}$

Step 5

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

$s' = t^{4} \sec^{2} (t) + 4 t^{3} \tan (t) - \frac{1}{2 t^{\frac{1}{2}}}$

Step 6

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

$\frac{d s}{d t} = t^{4} \sec^{2} (t) + 4 t^{3} \tan (t) - \frac{1}{2 t^{\frac{1}{2}}}$