Calculus Examples

$f (- \sqrt{6}) = \frac{12 x^{2}}{x^{4} + 36}$

Step 1

Replace the variable $x$ with $- \sqrt{6}$ in the expression.

$f (- \sqrt{6}) = \frac{12 {(- \sqrt{6})}^{2}}{{(- \sqrt{6})}^{4} + 36}$

Step 2

Simplify the result.

Tap for more steps...

Step 2.1

Simplify the numerator.

Tap for more steps...

Step 2.1.1

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \frac{12 {(- 1)}^{2} {\sqrt{6}}^{2}}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \sqrt{6}) = \frac{12 \cdot (1 {\sqrt{6}}^{2})}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (- \sqrt{6}) = \frac{12 \cdot (1 {(6^{\frac{1}{2}})}^{2})}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.3.2

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

$f (- \sqrt{6}) = \frac{12 \cdot 1 \cdot 6^{\frac{1}{2} \cdot 2}}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.3.3

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

$f (- \sqrt{6}) = \frac{12 \cdot 1 \cdot 6^{\frac{2}{2}}}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.3.4.1

Cancel the common factor.

$f (- \sqrt{6}) = \frac{12 \cdot 1 \cdot 6^{\frac{2}{2}}}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.3.4.2

Rewrite the expression.

$f (- \sqrt{6}) = \frac{12 \cdot 1 \cdot 6}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.3.5

Evaluate the exponent.

$f (- \sqrt{6}) = \frac{12 \cdot 1 \cdot 6}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.4

Combine exponents.

Tap for more steps...

Step 2.1.4.1

Multiply $12$ by $1$ .

$f (- \sqrt{6}) = \frac{12 \cdot 6}{{(- \sqrt{6})}^{4} + 36}$

Step 2.1.4.2

Multiply $12$ by $6$ .

$f (- \sqrt{6}) = \frac{72}{{(- \sqrt{6})}^{4} + 36}$

Step 2.2

Simplify the denominator.

Tap for more steps...

Step 2.2.1

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \frac{72}{{(- 1)}^{4} {\sqrt{6}}^{4} + 36}$

Step 2.2.2

Raise $- 1$ to the power of $4$ .

$f (- \sqrt{6}) = \frac{72}{1 {\sqrt{6}}^{4} + 36}$

Step 2.2.3

Multiply ${\sqrt{6}}^{4}$ by $1$ .

$f (- \sqrt{6}) = \frac{72}{{\sqrt{6}}^{4} + 36}$

Step 2.2.4

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

Tap for more steps...

Step 2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (- \sqrt{6}) = \frac{72}{{(6^{\frac{1}{2}})}^{4} + 36}$

Step 2.2.4.2

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

$f (- \sqrt{6}) = \frac{72}{6^{\frac{1}{2} \cdot 4} + 36}$

Step 2.2.4.3

Combine $\frac{1}{2}$ and $4$ .

$f (- \sqrt{6}) = \frac{72}{6^{\frac{4}{2}} + 36}$

Step 2.2.4.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.2.4.4.1

Factor $2$ out of $4$ .

$f (- \sqrt{6}) = \frac{72}{6^{\frac{2 \cdot 2}{2}} + 36}$

Step 2.2.4.4.2

Cancel the common factors.

Tap for more steps...

Step 2.2.4.4.2.1

Factor $2$ out of $2$ .

$f (- \sqrt{6}) = \frac{72}{6^{\frac{2 \cdot 2}{2 (1)}} + 36}$

Step 2.2.4.4.2.2

Cancel the common factor.

$f (- \sqrt{6}) = \frac{72}{6^{\frac{2 \cdot 2}{2 \cdot 1}} + 36}$

Step 2.2.4.4.2.3

Rewrite the expression.

$f (- \sqrt{6}) = \frac{72}{6^{\frac{2}{1}} + 36}$

Step 2.2.4.4.2.4

Divide $2$ by $1$ .

$f (- \sqrt{6}) = \frac{72}{6^{2} + 36}$

Step 2.2.5

Raise $6$ to the power of $2$ .

$f (- \sqrt{6}) = \frac{72}{36 + 36}$

Step 2.2.6

Add $36$ and $36$ .

$f (- \sqrt{6}) = \frac{72}{72}$

Step 2.3

Divide $72$ by $72$ .

$f (- \sqrt{6}) = 1$

Step 2.4

The final answer is $1$ .

$1$

Step 3