Calculus Examples

$f (x) = 7 x^{3} - 6 x^{2} + 1$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $7 x^{3} - 6 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.1.2

Evaluate $\frac{d}{d x} [7 x^{3}]$ .

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Step 1.1.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{3}$ with respect to $x$ is $7 \frac{d}{d x} [x^{3}]$ .

$7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$7 (3 x^{2}) + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.1.2.3

Multiply $3$ by $7$ .

$21 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 1.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$21 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 1.1.3.2

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

$21 x^{2} - 6 (2 x) + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $2$ by $- 6$ .

$21 x^{2} - 12 x + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$21 x^{2} - 12 x + 0$

Step 1.1.4.2

Add $21 x^{2} - 12 x$ and $0$ .

$f' (x) = 21 x^{2} - 12 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $21 x^{2} - 12 x$ .

$21 x^{2} - 12 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $21 x^{2} - 12 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$21 x^{2} - 12 x = 0$

Step 2.2

Factor $3 x$ out of $21 x^{2} - 12 x$ .

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Step 2.2.1

Factor $3 x$ out of $21 x^{2}$ .

$3 x (7 x) - 12 x = 0$

Step 2.2.2

Factor $3 x$ out of $- 12 x$ .

$3 x (7 x) + 3 x (- 4) = 0$

Step 2.2.3

Factor $3 x$ out of $3 x (7 x) + 3 x (- 4)$ .

$3 x (7 x - 4) = 0$

Step 2.3

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

$x = 0$

$7 x - 4 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $7 x - 4$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $7 x - 4$ equal to $0$ .

$7 x - 4 = 0$

Step 2.5.2

Solve $7 x - 4 = 0$ for $x$ .

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Step 2.5.2.1

Add $4$ to both sides of the equation.

$7 x = 4$

Step 2.5.2.2

Divide each term in $7 x = 4$ by $7$ and simplify.

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Step 2.5.2.2.1

Divide each term in $7 x = 4$ by $7$ .

$\frac{7 x}{7} = \frac{4}{7}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $7$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{4}{7}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{7}$

Step 2.6

The final solution is all the values that make $3 x (7 x - 4) = 0$ true.

$x = 0, \frac{4}{7}$

Step 3

Find the values where the derivative is undefined.

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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 $7 x^{3} - 6 x^{2} + 1$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$7 {(0)}^{3} - 6 {(0)}^{2} + 1$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$7 \cdot 0 - 6 {(0)}^{2} + 1$

Step 4.1.2.1.2

Multiply $7$ by $0$ .

$0 - 6 {(0)}^{2} + 1$

Step 4.1.2.1.3

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

$0 - 6 \cdot 0 + 1$

Step 4.1.2.1.4

Multiply $- 6$ by $0$ .

$0 + 0 + 1$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$0 + 1$

Step 4.1.2.2.2

Add $0$ and $1$ .

$1$

Step 4.2

Evaluate at $x = \frac{4}{7}$ .

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Step 4.2.1

Substitute $\frac{4}{7}$ for $x$ .

$7 {(\frac{4}{7})}^{3} - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Apply the product rule to $\frac{4}{7}$ .

$7 \frac{4^{3}}{7^{3}} - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2.1.2

Raise $4$ to the power of $3$ .

$7 \frac{64}{7^{3}} - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2.1.3

Raise $7$ to the power of $3$ .

$7 (\frac{64}{343}) - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2.1.4

Cancel the common factor of $7$ .

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Step 4.2.2.1.4.1

Factor $7$ out of $343$ .

$7 \frac{64}{7 (49)} - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2.1.4.2

Cancel the common factor.

$7 \frac{64}{7 \cdot 49} - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2.1.4.3

Rewrite the expression.

$\frac{64}{49} - 6 {(\frac{4}{7})}^{2} + 1$

Step 4.2.2.1.5

Apply the product rule to $\frac{4}{7}$ .

$\frac{64}{49} - 6 \frac{4^{2}}{7^{2}} + 1$

Step 4.2.2.1.6

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

$\frac{64}{49} - 6 \frac{16}{7^{2}} + 1$

Step 4.2.2.1.7

Raise $7$ to the power of $2$ .

$\frac{64}{49} - 6 (\frac{16}{49}) + 1$

Step 4.2.2.1.8

Multiply $- 6 (\frac{16}{49})$ .

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Step 4.2.2.1.8.1

Combine $- 6$ and $\frac{16}{49}$ .

$\frac{64}{49} + \frac{- 6 \cdot 16}{49} + 1$

Step 4.2.2.1.8.2

Multiply $- 6$ by $16$ .

$\frac{64}{49} + \frac{- 96}{49} + 1$

Step 4.2.2.1.9

Move the negative in front of the fraction.

$\frac{64}{49} - \frac{96}{49} + 1$

Step 4.2.2.2

Combine fractions.

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Step 4.2.2.2.1

Combine the numerators over the common denominator.

$1 + \frac{64 - 96}{49}$

Step 4.2.2.2.2

Simplify the expression.

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Step 4.2.2.2.2.1

Subtract $96$ from $64$ .

$1 + \frac{- 32}{49}$

Step 4.2.2.2.2.2

Move the negative in front of the fraction.

$1 - \frac{32}{49}$

Step 4.2.2.2.2.3

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

$\frac{49}{49} - \frac{32}{49}$

Step 4.2.2.2.2.4

Combine the numerators over the common denominator.

$\frac{49 - 32}{49}$

Step 4.2.2.2.2.5

Subtract $32$ from $49$ .

$\frac{17}{49}$

Step 4.3

List all of the points.

$(0, 1), (\frac{4}{7}, \frac{17}{49})$

Step 5