Calculus Examples

$y = \frac{x^{2}}{x^{2} + 3}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 3$ .

$f' (x) = \frac{(x^{2} + 3) \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{(x^{2} + 3) (2 x) - x^{2} \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.2

Move $2$ to the left of $x^{2} + 3$ .

$f' (x) = \frac{2 \cdot ((x^{2} + 3) x) - x^{2} \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.3

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$f' (x) = \frac{2 (x^{2} + 3) x - x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (3))}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{2 (x^{2} + 3) x - x^{2} (2 x + \frac{d}{d x} (3))}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.5

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x^{2} + 3) x - x^{2} (2 x + 0)}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.6

Simplify the expression.

Tap for more steps...

Step 1.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{2 (x^{2} + 3) x - x^{2} (2 x)}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{2 (x^{2} + 3) x - 2 x^{2} x}{{(x^{2} + 3)}^{2}}$

Step 1.1.3

Raise $x$ to the power of $1$ .

$f' (x) = \frac{2 (x^{2} + 3) x - 2 (x \cdot x^{2})}{{(x^{2} + 3)}^{2}}$

Step 1.1.4

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

$f' (x) = \frac{2 (x^{2} + 3) x - 2 x^{1 + 2}}{{(x^{2} + 3)}^{2}}$

Step 1.1.5

Add $1$ and $2$ .

$f' (x) = \frac{2 (x^{2} + 3) x - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6

Simplify.

Tap for more steps...

Step 1.1.6.1

Apply the distributive property.

$f' (x) = \frac{(2 x^{2} + 2 \cdot 3) x - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.2

Apply the distributive property.

$f' (x) = \frac{2 x^{2} x + 2 \cdot (3 x) - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3

Simplify the numerator.

Tap for more steps...

Step 1.1.6.3.1

Simplify each term.

Tap for more steps...

Step 1.1.6.3.1.1

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

Tap for more steps...

Step 1.1.6.3.1.1.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (3 x) - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3.1.1.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.1.6.3.1.1.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (3 x) - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3.1.1.2.2

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

$f' (x) = \frac{2 x^{1 + 2} + 2 \cdot (3 x) - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3.1.1.3

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} + 2 \cdot (3 x) - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3.1.2

Multiply $2$ by $3$ .

$f' (x) = \frac{2 x^{3} + 6 x - 2 x^{3}}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3.2

Combine the opposite terms in $2 x^{3} + 6 x - 2 x^{3}$ .

Tap for more steps...

Step 1.1.6.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = \frac{6 x + 0}{{(x^{2} + 3)}^{2}}$

Step 1.1.6.3.2.2

Add $6 x$ and $0$ .

$f' (x) = \frac{6 x}{{(x^{2} + 3)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{6 x}{{(x^{2} + 3)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{6 x}{{(x^{2} + 3)}^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{6 x}{{(x^{2} + 3)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$6 x = 0$

Step 2.3

Divide each term in $6 x = 0$ by $6$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $6 x = 0$ by $6$ .

$\frac{6 x}{6} = \frac{0}{6}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{0}{6}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{6}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $0$ by $6$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{x^{2}}{x^{2} + 3}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

$\frac{{(0)}^{2}}{{(0)}^{2} + 3}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0^{2} + 3}$

Step 4.1.2.2

Simplify the denominator.

Tap for more steps...

Step 4.1.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 + 3}$

Step 4.1.2.2.2

Add $0$ and $3$ .

$\frac{0}{3}$

Step 4.1.2.3

Divide $0$ by $3$ .

$0$

Step 4.2

List all of the points.

$(0, 0)$

Step 5