Calculus Examples

$f (x) = x^{6} + 5 x^{3} - x^{2} + 9$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{6} + 5 x^{3} - x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{6}] + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{6}] + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$6 x^{5} + \frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$

Step 1.2

Evaluate $\frac{d}{d x} [5 x^{3}]$ .

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Step 1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$6 x^{5} + 5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$

Step 1.2.2

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

$6 x^{5} + 5 (3 x^{2}) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$

Step 1.2.3

Multiply $3$ by $5$ .

$6 x^{5} + 15 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$

Step 1.3

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

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Step 1.3.1

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

$6 x^{5} + 15 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 1.3.2

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

$6 x^{5} + 15 x^{2} - (2 x) + \frac{d}{d x} [9]$

Step 1.3.3

Multiply $2$ by $- 1$ .

$6 x^{5} + 15 x^{2} - 2 x + \frac{d}{d x} [9]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$6 x^{5} + 15 x^{2} - 2 x + 0$

Step 1.4.2

Add $6 x^{5} + 15 x^{2} - 2 x$ and $0$ .

$6 x^{5} + 15 x^{2} - 2 x$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.39902337, 0, 0.13320739$

Step 3

Solve the original function $f (x) = x^{6} + 5 x^{3} - x^{2} + 9$ at $x = - 1.39902337$ .

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Step 3.1

Replace the variable $x$ with $- 1.39902337$ in the expression.

$f (- 1.39902337) = {(- 1.39902337)}^{6} + 5 {(- 1.39902337)}^{3} - {(- 1.39902337)}^{2} + 9$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 1.39902337$ to the power of $6$ .

$f (- 1.39902337) = 7.49807575 + 5 {(- 1.39902337)}^{3} - {(- 1.39902337)}^{2} + 9$

Step 3.2.1.2

Raise $- 1.39902337$ to the power of $3$ .

$f (- 1.39902337) = 7.49807575 + 5 \cdot - 2.73826144 - {(- 1.39902337)}^{2} + 9$

Step 3.2.1.3

Multiply $5$ by $- 2.73826144$ .

$f (- 1.39902337) = 7.49807575 - 13.69130723 - {(- 1.39902337)}^{2} + 9$

Step 3.2.1.4

Raise $- 1.39902337$ to the power of $2$ .

$f (- 1.39902337) = 7.49807575 - 13.69130723 - 1 \cdot 1.9572664 + 9$

Step 3.2.1.5

Multiply $- 1$ by $1.9572664$ .

$f (- 1.39902337) = 7.49807575 - 13.69130723 - 1.9572664 + 9$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $13.69130723$ from $7.49807575$ .

$f (- 1.39902337) = - 6.19323148 - 1.9572664 + 9$

Step 3.2.2.2

Subtract $1.9572664$ from $- 6.19323148$ .

$f (- 1.39902337) = - 8.15049788 + 9$

Step 3.2.2.3

Add $- 8.15049788$ and $9$ .

$f (- 1.39902337) = 0.84950211$

Step 3.2.3

The final answer is $0.84950211$ .

$0.84950211$

Step 4

Solve the original function $f (x) = x^{6} + 5 x^{3} - x^{2} + 9$ at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{6} + 5 {(0)}^{3} - {(0)}^{2} + 9$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f (0) = 0 + 5 {(0)}^{3} - {(0)}^{2} + 9$

Step 4.2.1.2

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

$f (0) = 0 + 5 \cdot 0 - {(0)}^{2} + 9$

Step 4.2.1.3

Multiply $5$ by $0$ .

$f (0) = 0 + 0 - {(0)}^{2} + 9$

Step 4.2.1.4

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

$f (0) = 0 + 0 - 0 + 9$

Step 4.2.1.5

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0 + 0 + 9$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 9$

Step 4.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 9$

Step 4.2.2.3

Add $0$ and $9$ .

$f (0) = 9$

Step 4.2.3

The final answer is $9$ .

$9$

Step 5

Solve the original function $f (x) = x^{6} + 5 x^{3} - x^{2} + 9$ at $x = 0.13320739$ .

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Step 5.1

Replace the variable $x$ with $0.13320739$ in the expression.

$f (0.13320739) = {(0.13320739)}^{6} + 5 {(0.13320739)}^{3} - {(0.13320739)}^{2} + 9$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $0.13320739$ to the power of $6$ .

$f (0.13320739) = 0.00000558 + 5 {(0.13320739)}^{3} - {(0.13320739)}^{2} + 9$

Step 5.2.1.2

Raise $0.13320739$ to the power of $3$ .

$f (0.13320739) = 0.00000558 + 5 \cdot 0.00236365 - {(0.13320739)}^{2} + 9$

Step 5.2.1.3

Multiply $5$ by $0.00236365$ .

$f (0.13320739) = 0.00000558 + 0.01181829 - {(0.13320739)}^{2} + 9$

Step 5.2.1.4

Raise $0.13320739$ to the power of $2$ .

$f (0.13320739) = 0.00000558 + 0.01181829 - 1 \cdot 0.0177442 + 9$

Step 5.2.1.5

Multiply $- 1$ by $0.0177442$ .

$f (0.13320739) = 0.00000558 + 0.01181829 - 0.0177442 + 9$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $0.00000558$ and $0.01181829$ .

$f (0.13320739) = 0.01182388 - 0.0177442 + 9$

Step 5.2.2.2

Subtract $0.0177442$ from $0.01182388$ .

$f (0.13320739) = - 0.00592032 + 9$

Step 5.2.2.3

Add $- 0.00592032$ and $9$ .

$f (0.13320739) = 8.99407967$

Step 5.2.3

The final answer is $8.99407967$ .

$8.99407967$

Step 6

The horizontal tangent lines on function $f (x) = x^{6} + 5 x^{3} - x^{2} + 9$ are $y = 0.84950211, y = 9, y = 8.99407967$ .

$y = 0.84950211, y = 9, y = 8.99407967$

Step 7