Calculus Examples

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

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

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

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

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

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

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

Multiply $4$ by $4$ .

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

Reorder terms.

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

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

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

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

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

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

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Rewrite the equation as $16 x^{3} - 2 x + 1 = 0$ .

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

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

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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.

$p = \pm 1$

$q = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 = 0$

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

$\pm 1, \pm 0.5, \pm 0.25, \pm 0.125, \pm 0.0625 = 0$

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.

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Substitute $- 0.5$ into the polynomial.

$16 {(- 0.5)}^{3} - 2 \cdot - 0.5 + 1 = 0$

Simplify each term.

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

$16 \cdot - 0.125 - 2 \cdot - 0.5 + 1 = 0$

Multiply $16$ by $- 0.125$ .

$- 2 - 2 \cdot - 0.5 + 1 = 0$

Multiply $- 2$ by $- 0.5$ .

$- 2 + 1 + 1 = 0$

Add $- 2$ and $1$ .

$- 1 + 1 = 0$

Add $- 1$ and $1$ .

$0 = 0$

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.

$\frac{16 x^{3} - 2 x + 1}{2 x + 1} = 0$

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

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

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

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

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

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Set the factor equal to $0$ .

$2 x + 1 = 0$

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term by $2$ and simplify.

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Move the negative in front of the fraction.

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

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

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Set the factor equal to $0$ .

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

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Simplify the numerator.

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

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

Multiply $8$ by $1$ .

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

Multiply $- 4$ by $8$ .

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

Subtract $32$ from $16$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $8$ .

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

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

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

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

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Simplify the numerator.

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

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

Multiply $8$ by $1$ .

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

Multiply $- 4$ by $8$ .

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

Subtract $32$ from $16$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $8$ .

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

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

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

Change the $\pm$ to $+$ .

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

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

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

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

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Simplify the numerator.

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

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

Multiply $8$ by $1$ .

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

Multiply $- 4$ by $8$ .

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

Subtract $32$ from $16$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $8$ .

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

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

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

Change the $\pm$ to $-$ .

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

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

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

Move the negative in front of the fraction.

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

The final answer is the combination of both solutions.

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

The solution is the result of $2 x + 1 = 0$ and $8 x^{2} - 4 x + 1 = 0$ .

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

Solve for $y$ .

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Remove the parentheses around the expression $- \frac{1}{2}$ .

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

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

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Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

One to any power is one.

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

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

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

Cancel the common factor of $4$ .

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Write $4$ as a fraction with denominator $1$ .

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

Factor out the greatest common factor $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{1}$ and $\frac{1}{4}$ .

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

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $1$ from $- 2$ .

$y = \frac{- 3}{4} + \frac{1}{4}$

Move the negative in front of the fraction.

$y = - \frac{3}{4} + \frac{1}{4}$

Combine the numerators over the common denominator.

$y = \frac{- 3 + 1}{4}$

Add $- 3$ and $1$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $- 2$ .

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

Cancel the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Calculated $x$ values cannot contain imaginary components.

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

Calculated $x$ values cannot contain imaginary components.

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

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

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

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