Calculus Examples

$q = \cos (\frac{t}{\sqrt{t + 2}})$

Step 1

Multiply $\frac{t}{\sqrt{t + 2}}$ by $\frac{\sqrt{t + 2}}{\sqrt{t + 2}}$ .

$q = \cos (\frac{t}{\sqrt{t + 2}} \cdot \frac{\sqrt{t + 2}}{\sqrt{t + 2}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{t}{\sqrt{t + 2}}$ by $\frac{\sqrt{t + 2}}{\sqrt{t + 2}}$ .

$q = \cos (\frac{t \sqrt{t + 2}}{\sqrt{t + 2} \sqrt{t + 2}})$

Step 2.2

Raise $\sqrt{t + 2}$ to the power of $1$ .

$q = \cos (\frac{t \sqrt{t + 2}}{{\sqrt{t + 2}}^{1} \sqrt{t + 2}})$

Step 2.3

Raise $\sqrt{t + 2}$ to the power of $1$ .

$q = \cos (\frac{t \sqrt{t + 2}}{{\sqrt{t + 2}}^{1} {\sqrt{t + 2}}^{1}})$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$q = \cos (\frac{t \sqrt{t + 2}}{{\sqrt{t + 2}}^{1 + 1}})$

Step 2.5

Add $1$ and $1$ .

$q = \cos (\frac{t \sqrt{t + 2}}{{\sqrt{t + 2}}^{2}})$

Step 2.6

Rewrite ${\sqrt{t + 2}}^{2}$ as $t + 2$ .

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Step 2.6.1

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

$q = \cos (\frac{t \sqrt{t + 2}}{{({(t + 2)}^{\frac{1}{2}})}^{2}})$

Step 2.6.2

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

$q = \cos (\frac{t \sqrt{t + 2}}{{(t + 2)}^{\frac{1}{2} \cdot 2}})$

Step 2.6.3

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

$q = \cos (\frac{t \sqrt{t + 2}}{{(t + 2)}^{\frac{2}{2}}})$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$q = \cos (\frac{t \sqrt{t + 2}}{{(t + 2)}^{\frac{2}{2}}})$

Step 2.6.4.2

Rewrite the expression.

$q = \cos (\frac{t \sqrt{t + 2}}{{(t + 2)}^{1}})$

Step 2.6.5

Simplify.

$q = \cos (\frac{t \sqrt{t + 2}}{t + 2})$