Calculus Examples

$y = x - x^{2} + 4 x^{4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x - x^{2} + 4 x^{4})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $x - x^{2} + 4 x^{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4 x^{4}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4 x^{4}]$

Step 3.1.2

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

$1 + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4 x^{4}]$

Step 3.2

Evaluate $\frac{d}{d x} [- x^{2}]$ .

Tap for more steps...

Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$1 - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x^{4}]$

Step 3.2.2

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

$1 - (2 x) + \frac{d}{d x} [4 x^{4}]$

Step 3.2.3

Multiply $2$ by $- 1$ .

$1 - 2 x + \frac{d}{d x} [4 x^{4}]$

Step 3.3

Evaluate $\frac{d}{d x} [4 x^{4}]$ .

Tap for more steps...

Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{4}$ with respect to $x$ is $4 \frac{d}{d x} [x^{4}]$ .

$1 - 2 x + 4 \frac{d}{d x} [x^{4}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$1 - 2 x + 4 (4 x^{3})$

Step 3.3.3

Multiply $4$ by $4$ .

$1 - 2 x + 16 x^{3}$

Step 3.4

Reorder terms.

$16 x^{3} - 2 x + 1$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 16 x^{3} - 2 x + 1$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 16 x^{3} - 2 x + 1$

Step 6

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

Tap for more steps...

Step 6.1

Factor $16 x^{3} - 2 x + 1$ using the rational roots test.

Tap for more steps...

Step 6.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

Step 6.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.0625, \pm 0.5, \pm 0.125, \pm 0.25$

Step 6.1.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Tap for more steps...

Step 6.1.3.1

Substitute $- 0.5$ into the polynomial.

$16 {(- 0.5)}^{3} - 2 \cdot - 0.5 + 1$

Step 6.1.3.2

Raise $- 0.5$ to the power of $3$ .

$16 \cdot - 0.125 - 2 \cdot - 0.5 + 1$

Step 6.1.3.3

Multiply $16$ by $- 0.125$ .

$- 2 - 2 \cdot - 0.5 + 1$

Step 6.1.3.4

Multiply $- 2$ by $- 0.5$ .

$- 2 + 1 + 1$

Step 6.1.3.5

Add $- 2$ and $1$ .

$- 1 + 1$

Step 6.1.3.6

Add $- 1$ and $1$ .

$0$

Step 6.1.4

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{16 x^{3} - 2 x + 1}{2 x + 1}$

Step 6.1.5

Divide $16 x^{3} - 2 x + 1$ by $2 x + 1$ .

Tap for more steps...

Step 6.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$16 x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 6.1.5.2

Divide the highest order term in the dividend $16 x^{3}$ by the highest order term in divisor $2 x$ .

					$8 x^{2}$
	$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$

Step 6.1.5.3

Multiply the new quotient term by the divisor.

				$8 x^{2}$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			+	$16 x^{3}$	+	$8 x^{2}$

Step 6.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{3} + 8 x^{2}$

				$8 x^{2}$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$

Step 6.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$8 x^{2}$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$

Step 6.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$8 x^{2}$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$

Step 6.1.5.7

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $2 x$ .

				$8 x^{2}$	-	$4 x$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$

Step 6.1.5.8

Multiply the new quotient term by the divisor.

				$8 x^{2}$	-	$4 x$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					-	$8 x^{2}$	-	$4 x$

Step 6.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 4 x$

				$8 x^{2}$	-	$4 x$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$

Step 6.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$8 x^{2}$	-	$4 x$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$
							+	$2 x$

Step 6.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$8 x^{2}$	-	$4 x$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$
							+	$2 x$	+	$1$

Step 6.1.5.12

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $2 x$ .

				$8 x^{2}$	-	$4 x$	+	$1$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$
							+	$2 x$	+	$1$

Step 6.1.5.13

Multiply the new quotient term by the divisor.

				$8 x^{2}$	-	$4 x$	+	$1$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$
							+	$2 x$	+	$1$
							+	$2 x$	+	$1$

Step 6.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 1$

				$8 x^{2}$	-	$4 x$	+	$1$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$
							+	$2 x$	+	$1$
							-	$2 x$	-	$1$

Step 6.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$8 x^{2}$	-	$4 x$	+	$1$
$2 x$	+	$1$		$16 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$1$
			-	$16 x^{3}$	-	$8 x^{2}$
					-	$8 x^{2}$	-	$2 x$
					+	$8 x^{2}$	+	$4 x$
							+	$2 x$	+	$1$
							-	$2 x$	-	$1$
										$0$

Step 6.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$8 x^{2} - 4 x + 1$

Step 6.1.6

Write $16 x^{3} - 2 x + 1$ as a set of factors.

$(2 x + 1) (8 x^{2} - 4 x + 1) = 0$

Step 6.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 1 = 0$

$8 x^{2} - 4 x + 1 = 0$

Step 6.3

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 6.3.2

Solve $2 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 6.3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 6.3.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

Tap for more steps...

Step 6.3.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 6.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 6.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 6.3.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 6.4

Set $8 x^{2} - 4 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.4.1

Set $8 x^{2} - 4 x + 1$ equal to $0$ .

$8 x^{2} - 4 x + 1 = 0$

Step 6.4.2

Solve $8 x^{2} - 4 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 6.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.4.2.2

Substitute the values $a = 8$ , $b = - 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (8 \cdot 1)}}{2 \cdot 8}$

Step 6.4.2.3

Simplify.

Tap for more steps...

Step 6.4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 6.4.2.3.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 8 \cdot 1}}{2 \cdot 8}$

Step 6.4.2.3.1.2

Multiply $- 4 \cdot 8 \cdot 1$ .

Tap for more steps...

Step 6.4.2.3.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{4 \pm \sqrt{16 - 32 \cdot 1}}{2 \cdot 8}$

Step 6.4.2.3.1.2.2

Multiply $- 32$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 32}}{2 \cdot 8}$

Step 6.4.2.3.1.3

Subtract $32$ from $16$ .

$x = \frac{4 \pm \sqrt{- 16}}{2 \cdot 8}$

Step 6.4.2.3.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 8}$

Step 6.4.2.3.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 8}$

Step 6.4.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{16}}{2 \cdot 8}$

Step 6.4.2.3.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 8}$

Step 6.4.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 4}{2 \cdot 8}$

Step 6.4.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{4 \pm 4 i}{2 \cdot 8}$

Step 6.4.2.3.2

Multiply $2$ by $8$ .

$x = \frac{4 \pm 4 i}{16}$

Step 6.4.2.3.3

Simplify $\frac{4 \pm 4 i}{16}$ .

$x = \frac{1 \pm i}{4}$

Step 6.4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 6.4.2.4.1

Simplify the numerator.

Tap for more steps...

Step 6.4.2.4.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 8 \cdot 1}}{2 \cdot 8}$

Step 6.4.2.4.1.2

Multiply $- 4 \cdot 8 \cdot 1$ .

Tap for more steps...

Step 6.4.2.4.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{4 \pm \sqrt{16 - 32 \cdot 1}}{2 \cdot 8}$

Step 6.4.2.4.1.2.2

Multiply $- 32$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 32}}{2 \cdot 8}$

Step 6.4.2.4.1.3

Subtract $32$ from $16$ .

$x = \frac{4 \pm \sqrt{- 16}}{2 \cdot 8}$

Step 6.4.2.4.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 8}$

Step 6.4.2.4.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 8}$

Step 6.4.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{16}}{2 \cdot 8}$

Step 6.4.2.4.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 8}$

Step 6.4.2.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 4}{2 \cdot 8}$

Step 6.4.2.4.1.9

Move $4$ to the left of $i$ .

$x = \frac{4 \pm 4 i}{2 \cdot 8}$

Step 6.4.2.4.2

Multiply $2$ by $8$ .

$x = \frac{4 \pm 4 i}{16}$

Step 6.4.2.4.3

Simplify $\frac{4 \pm 4 i}{16}$ .

$x = \frac{1 \pm i}{4}$

Step 6.4.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + i}{4}$

Step 6.4.2.4.5

Split the fraction $\frac{1 + i}{4}$ into two fractions.

$x = \frac{1}{4} + \frac{i}{4}$

Step 6.4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 6.4.2.5.1

Simplify the numerator.

Tap for more steps...

Step 6.4.2.5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 8 \cdot 1}}{2 \cdot 8}$

Step 6.4.2.5.1.2

Multiply $- 4 \cdot 8 \cdot 1$ .

Tap for more steps...

Step 6.4.2.5.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{4 \pm \sqrt{16 - 32 \cdot 1}}{2 \cdot 8}$

Step 6.4.2.5.1.2.2

Multiply $- 32$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 32}}{2 \cdot 8}$

Step 6.4.2.5.1.3

Subtract $32$ from $16$ .

$x = \frac{4 \pm \sqrt{- 16}}{2 \cdot 8}$

Step 6.4.2.5.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 8}$

Step 6.4.2.5.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 8}$

Step 6.4.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{16}}{2 \cdot 8}$

Step 6.4.2.5.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 8}$

Step 6.4.2.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 4}{2 \cdot 8}$

Step 6.4.2.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{4 \pm 4 i}{2 \cdot 8}$

Step 6.4.2.5.2

Multiply $2$ by $8$ .

$x = \frac{4 \pm 4 i}{16}$

Step 6.4.2.5.3

Simplify $\frac{4 \pm 4 i}{16}$ .

$x = \frac{1 \pm i}{4}$

Step 6.4.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - i}{4}$

Step 6.4.2.5.5

Split the fraction $\frac{1 - i}{4}$ into two fractions.

$x = \frac{1}{4} + \frac{- i}{4}$

Step 6.4.2.5.6

Move the negative in front of the fraction.

$x = \frac{1}{4} - \frac{i}{4}$

Step 6.4.2.6

The final answer is the combination of both solutions.

$x = \frac{1}{4} + \frac{i}{4}, \frac{1}{4} - \frac{i}{4}$

Step 6.5

The final solution is all the values that make $(2 x + 1) (8 x^{2} - 4 x + 1) = 0$ true.

$x = - \frac{1}{2}, \frac{1}{4} + \frac{i}{4}, \frac{1}{4} - \frac{i}{4}$

Step 7

Solve for $y$ .

Tap for more steps...

Step 7.1

Remove parentheses.

$y = - \frac{1}{2} - {(- \frac{1}{2})}^{2} + 4 {(- \frac{1}{2})}^{4}$

Step 7.2

Remove parentheses.

$y = (- \frac{1}{2}) - {(- \frac{1}{2})}^{2} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3

Simplify $(- \frac{1}{2}) - {(- \frac{1}{2})}^{2} + 4 {(- \frac{1}{2})}^{4}$ .

Tap for more steps...

Step 7.3.1

Simplify each term.

Tap for more steps...

Step 7.3.1.1

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

Tap for more steps...

Step 7.3.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

$y = - \frac{1}{2} - ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.1.2

Apply the product rule to $\frac{1}{2}$ .

$y = - \frac{1}{2} - ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 7.3.1.2.1

Move ${(- 1)}^{2}$ .

$y = - \frac{1}{2} + {(- 1)}^{2} \cdot - 1 \frac{1^{2}}{2^{2}} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

Tap for more steps...

Step 7.3.1.2.2.1

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

$y = - \frac{1}{2} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{1^{2}}{2^{2}} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.2.2.2

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

$y = - \frac{1}{2} + {(- 1)}^{2 + 1} \frac{1^{2}}{2^{2}} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.2.3

Add $2$ and $1$ .

$y = - \frac{1}{2} + {(- 1)}^{3} \frac{1^{2}}{2^{2}} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.3

Raise $- 1$ to the power of $3$ .

$y = - \frac{1}{2} - \frac{1^{2}}{2^{2}} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.4

One to any power is one.

$y = - \frac{1}{2} - \frac{1}{2^{2}} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.5

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

$y = - \frac{1}{2} - \frac{1}{4} + 4 {(- \frac{1}{2})}^{4}$

Step 7.3.1.6

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

Tap for more steps...

Step 7.3.1.6.1

Apply the product rule to $- \frac{1}{2}$ .

$y = - \frac{1}{2} - \frac{1}{4} + 4 ({(- 1)}^{4} {(\frac{1}{2})}^{4})$

Step 7.3.1.6.2

Apply the product rule to $\frac{1}{2}$ .

$y = - \frac{1}{2} - \frac{1}{4} + 4 ({(- 1)}^{4} \frac{1^{4}}{2^{4}})$

Step 7.3.1.7

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

$y = - \frac{1}{2} - \frac{1}{4} + 4 (1 \frac{1^{4}}{2^{4}})$

Step 7.3.1.8

Multiply $\frac{1^{4}}{2^{4}}$ by $1$ .

$y = - \frac{1}{2} - \frac{1}{4} + 4 \frac{1^{4}}{2^{4}}$

Step 7.3.1.9

One to any power is one.

$y = - \frac{1}{2} - \frac{1}{4} + 4 \frac{1}{2^{4}}$

Step 7.3.1.10

Raise $2$ to the power of $4$ .

$y = - \frac{1}{2} - \frac{1}{4} + 4 (\frac{1}{16})$

Step 7.3.1.11

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.3.1.11.1

Factor $4$ out of $16$ .

$y = - \frac{1}{2} - \frac{1}{4} + 4 \frac{1}{4 (4)}$

Step 7.3.1.11.2

Cancel the common factor.

$y = - \frac{1}{2} - \frac{1}{4} + 4 \frac{1}{4 \cdot 4}$

Step 7.3.1.11.3

Rewrite the expression.

$y = - \frac{1}{2} - \frac{1}{4} + \frac{1}{4}$

Step 7.3.2

Combine fractions.

Tap for more steps...

Step 7.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 1}{2} + \frac{- 1 + 1}{4}$

Step 7.3.2.2

Add $- 1$ and $1$ .

$y = \frac{- 1}{2} + \frac{0}{4}$

Step 7.3.3

Simplify each term.

Tap for more steps...

Step 7.3.3.1

Move the negative in front of the fraction.

$y = - \frac{1}{2} + \frac{0}{4}$

Step 7.3.3.2

Divide $0$ by $4$ .

$y = - \frac{1}{2} + 0$

Step 7.3.4

Add $- \frac{1}{2}$ and $0$ .

$y = - \frac{1}{2}$

Step 8

Calculated $x$ values cannot contain imaginary components.

$\frac{1}{4} + \frac{i}{4}$ is not a valid value for x

Step 9

Calculated $x$ values cannot contain imaginary components.

$\frac{1}{4} - \frac{i}{4}$ is not a valid value for x

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(- \frac{1}{2}, - \frac{1}{2})$

Step 11

Enter YOUR Problem