Calculus Examples

$f'' (t) = \frac{3}{\sqrt{t}}$

Step 1

The function $f' (t)$ can be found by evaluating the indefinite integral of the derivative $f'' (t)$ .

$f' (t) = \int f'' (t) d t$

Step 2

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int \frac{1}{\sqrt{t}} d t$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$3 \int \frac{1}{t^{\frac{1}{2}}} d t$

Step 3.2

Move $t^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$3 \int {(t^{\frac{1}{2}})}^{- 1} d t$

Step 3.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{- 1}$ .

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Step 3.3.1

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

$3 \int t^{\frac{1}{2} \cdot - 1} d t$

Step 3.3.2

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

$3 \int t^{\frac{- 1}{2}} d t$

Step 3.3.3

Move the negative in front of the fraction.

$3 \int t^{- \frac{1}{2}} d t$

Step 4

By the Power Rule, the integral of $t^{- \frac{1}{2}}$ with respect to $t$ is $2 t^{\frac{1}{2}}$ .

$3 (2 t^{\frac{1}{2}} + C)$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $3 (2 t^{\frac{1}{2}} + C)$ as $3 \cdot 2 t^{\frac{1}{2}} + C$ .

$3 \cdot 2 t^{\frac{1}{2}} + C$

Step 5.2

Multiply $3$ by $2$ .

$6 t^{\frac{1}{2}} + C$

Step 6

The function $f'$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f' (t) = 6 t^{\frac{1}{2}} + C$

Step 7

The function $f (t)$ can be found by evaluating the indefinite integral of the derivative $f' (t)$ .

$f (t) = \int f' (t) d t$

Step 8

Split the single integral into multiple integrals.

$\int 6 t^{\frac{1}{2}} d t + \int C d t$

Step 9

Since $6$ is constant with respect to $t$ , move $6$ out of the integral.

$6 \int t^{\frac{1}{2}} d t + \int C d t$

Step 10

By the Power Rule, the integral of $t^{\frac{1}{2}}$ with respect to $t$ is $\frac{2}{3} t^{\frac{3}{2}}$ .

$6 (\frac{2}{3} t^{\frac{3}{2}} + C) + \int C d t$

Step 11

Apply the constant rule.

$6 (\frac{2}{3} t^{\frac{3}{2}} + C) + C t + C$

Step 12

Simplify.

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Step 12.1

Combine $\frac{2}{3}$ and $t^{\frac{3}{2}}$ .

$6 (\frac{2 t^{\frac{3}{2}}}{3} + C) + C t + C$

Step 12.2

Simplify.

$4 t^{\frac{3}{2}} + C t + C$

Step 13

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f (t) = 4 t^{\frac{3}{2}} + C t + C$