Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

Step 3.2.2

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

$1 + 4 (2 x)$

Step 3.2.3

Multiply $2$ by $4$ .

$1 + 8 x$

Step 3.3

Reorder terms.

$8 x + 1$

Step 4

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

$y' = 8 x + 1$

Step 5

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

$\frac{d y}{d x} = 8 x + 1$

Step 6

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

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Step 6.1

Subtract $1$ from both sides of the equation.

$8 x = - 1$

Step 6.2

Divide each term in $8 x = - 1$ by $8$ and simplify.

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Step 6.2.1

Divide each term in $8 x = - 1$ by $8$ .

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $8$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{8}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{8}$

Step 7

Solve for $y$ .

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Step 7.1

Remove parentheses.

$y = - \frac{1}{8} + 4 {(- \frac{1}{8})}^{2}$

Step 7.2

Remove parentheses.

$y = (- \frac{1}{8}) + 4 {(- \frac{1}{8})}^{2}$

Step 7.3

Simplify $(- \frac{1}{8}) + 4 {(- \frac{1}{8})}^{2}$ .

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

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

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Step 7.3.1.1.1

Apply the product rule to $- \frac{1}{8}$ .

$y = - \frac{1}{8} + 4 ({(- 1)}^{2} {(\frac{1}{8})}^{2})$

Step 7.3.1.1.2

Apply the product rule to $\frac{1}{8}$ .

$y = - \frac{1}{8} + 4 ({(- 1)}^{2} \frac{1^{2}}{8^{2}})$

Step 7.3.1.2

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

$y = - \frac{1}{8} + 4 (1 \frac{1^{2}}{8^{2}})$

Step 7.3.1.3

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

$y = - \frac{1}{8} + 4 \frac{1^{2}}{8^{2}}$

Step 7.3.1.4

One to any power is one.

$y = - \frac{1}{8} + 4 \frac{1}{8^{2}}$

Step 7.3.1.5

Raise $8$ to the power of $2$ .

$y = - \frac{1}{8} + 4 (\frac{1}{64})$

Step 7.3.1.6

Cancel the common factor of $4$ .

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Step 7.3.1.6.1

Factor $4$ out of $64$ .

$y = - \frac{1}{8} + 4 \frac{1}{4 (16)}$

Step 7.3.1.6.2

Cancel the common factor.

$y = - \frac{1}{8} + 4 \frac{1}{4 \cdot 16}$

Step 7.3.1.6.3

Rewrite the expression.

$y = - \frac{1}{8} + \frac{1}{16}$

Step 7.3.2

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{1}{8} \cdot \frac{2}{2} + \frac{1}{16}$

Step 7.3.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.3.1

Multiply $\frac{1}{8}$ by $\frac{2}{2}$ .

$y = - \frac{2}{8 \cdot 2} + \frac{1}{16}$

Step 7.3.3.2

Multiply $8$ by $2$ .

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

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Add $- 2$ and $1$ .

$y = \frac{- 1}{16}$

Step 7.3.6

Move the negative in front of the fraction.

$y = - \frac{1}{16}$

Step 8

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

$(- \frac{1}{8}, - \frac{1}{16})$

Step 9

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