Calculus Examples

${(2 x {(x^{2} + 1)}^{\frac{1}{2}})}^{2}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $2 x {(x^{2} + 1)}^{\frac{1}{2}}$ .

${(2 x)}^{2} {({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$

Step 1.2

Apply the product rule to $2 x$ .

$2^{2} x^{2} {({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$

Step 2

Raise $2$ to the power of $2$ .

$4 x^{2} {({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$

Step 3

Multiply the exponents in ${({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

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

$4 x^{2} {(x^{2} + 1)}^{\frac{1}{2} \cdot 2}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$4 x^{2} {(x^{2} + 1)}^{\frac{1}{2} \cdot 2}$

Step 3.2.2

Rewrite the expression.

$4 x^{2} {(x^{2} + 1)}^{1}$

Step 4

Simplify.

$4 x^{2} (x^{2} + 1)$

Step 5

Apply the distributive property.

$4 x^{2} x^{2} + 4 x^{2} \cdot 1$

Step 6

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 6.1

Move $x^{2}$ .

$4 (x^{2} x^{2}) + 4 x^{2} \cdot 1$

Step 6.2

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

$4 x^{2 + 2} + 4 x^{2} \cdot 1$

Step 6.3

Add $2$ and $2$ .

$4 x^{4} + 4 x^{2} \cdot 1$

Step 7

Multiply $4$ by $1$ .

$4 x^{4} + 4 x^{2}$