Calculus Examples

$\sqrt[4]{t} - 4 e^{t}$

Step 1

By the Sum Rule, the derivative of $\sqrt[4]{t} - 4 e^{t}$ with respect to $t$ is $\frac{d}{d t} [\sqrt[4]{t}] + \frac{d}{d t} [- 4 e^{t}]$ .

$\frac{d}{d t} [\sqrt[4]{t}] + \frac{d}{d t} [- 4 e^{t}]$

Step 2

Evaluate $\frac{d}{d t} [\sqrt[4]{t}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{t}$ as $t^{\frac{1}{4}}$ .

$\frac{d}{d t} [t^{\frac{1}{4}}] + \frac{d}{d t} [- 4 e^{t}]$

Step 2.2

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

$\frac{1}{4} t^{\frac{1}{4} - 1} + \frac{d}{d t} [- 4 e^{t}]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} t^{\frac{1}{4} - 1 \cdot \frac{4}{4}} + \frac{d}{d t} [- 4 e^{t}]$

Step 2.4

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{1}{4} t^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}} + \frac{d}{d t} [- 4 e^{t}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{4} t^{\frac{1 - 1 \cdot 4}{4}} + \frac{d}{d t} [- 4 e^{t}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $4$ .

$\frac{1}{4} t^{\frac{1 - 4}{4}} + \frac{d}{d t} [- 4 e^{t}]$

Step 2.6.2

Subtract $4$ from $1$ .

$\frac{1}{4} t^{\frac{- 3}{4}} + \frac{d}{d t} [- 4 e^{t}]$

Step 2.7

Move the negative in front of the fraction.

$\frac{1}{4} t^{- \frac{3}{4}} + \frac{d}{d t} [- 4 e^{t}]$

Step 3

Evaluate $\frac{d}{d t} [- 4 e^{t}]$ .

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Step 3.1

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4 e^{t}$ with respect to $t$ is $- 4 \frac{d}{d t} [e^{t}]$ .

$\frac{1}{4} t^{- \frac{3}{4}} - 4 \frac{d}{d t} [e^{t}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $e$ .

$\frac{1}{4} t^{- \frac{3}{4}} - 4 e^{t}$

Step 4

Simplify.

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Step 4.1

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

$\frac{1}{4} \cdot \frac{1}{t^{\frac{3}{4}}} - 4 e^{t}$

Step 4.2

Multiply $\frac{1}{4}$ by $\frac{1}{t^{\frac{3}{4}}}$ .

$\frac{1}{4 t^{\frac{3}{4}}} - 4 e^{t}$