Calculus Examples

$y = 5 x^{6} - 3 x^{4} + 2 x - 9$

Step 1

Write $y = 5 x^{6} - 3 x^{4} + 2 x - 9$ as a function.

$f (x) = 5 x^{6} - 3 x^{4} + 2 x - 9$

Step 2

Find the first derivative.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{6} - 3 x^{4} + 2 x - 9$ with respect to $x$ is $\frac{d}{d x} [5 x^{6}] + \frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [5 x^{6}] + \frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{6}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{6}$ with respect to $x$ is $5 \frac{d}{d x} [x^{6}]$ .

$5 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.2.2

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

$5 (6 x^{5}) + \frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.2.3

Multiply $6$ by $5$ .

$30 x^{5} + \frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.3

Evaluate $\frac{d}{d x} [- 3 x^{4}]$ .

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Step 2.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

$30 x^{5} - 3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.3.2

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

$30 x^{5} - 3 (4 x^{3}) + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.3.3

Multiply $4$ by $- 3$ .

$30 x^{5} - 12 x^{3} + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$

Step 2.4

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

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Step 2.4.1

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

$30 x^{5} - 12 x^{3} + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

Step 2.4.2

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

$30 x^{5} - 12 x^{3} + 2 \cdot 1 + \frac{d}{d x} [- 9]$

Step 2.4.3

Multiply $2$ by $1$ .

$30 x^{5} - 12 x^{3} + 2 + \frac{d}{d x} [- 9]$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

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

$30 x^{5} - 12 x^{3} + 2 + 0$

Step 2.5.2

Add $30 x^{5} - 12 x^{3} + 2$ and $0$ .

$30 x^{5} - 12 x^{3} + 2$

Step 3

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

$x \approx - 0.74790241$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 0.74790241) \cup (- 0.74790241, \infty)$

Step 5

Substitute any number, such as $- 3$ , from the interval $(- \infty, - 0.74790241)$ in the first derivative $30 x^{5} - 12 x^{3} + 2$ to check if the result is negative or positive.

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Step 5.1

Replace the variable $x$ with $- 3$ in the expression.

$f' (- 3) = 30 {(- 3)}^{5} - 12 {(- 3)}^{3} + 2$

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 $- 3$ to the power of $5$ .

$f' (- 3) = 30 \cdot - 243 - 12 {(- 3)}^{3} + 2$

Step 5.2.1.2

Multiply $30$ by $- 243$ .

$f' (- 3) = - 7290 - 12 {(- 3)}^{3} + 2$

Step 5.2.1.3

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

$f' (- 3) = - 7290 - 12 \cdot - 27 + 2$

Step 5.2.1.4

Multiply $- 12$ by $- 27$ .

$f' (- 3) = - 7290 + 324 + 2$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 7290$ and $324$ .

$f' (- 3) = - 6966 + 2$

Step 5.2.2.2

Add $- 6966$ and $2$ .

$f' (- 3) = - 6964$

Step 5.2.3

The final answer is $- 6964$ .

$- 6964$

Step 6

Substitute any number, such as $0$ , from the interval $(- 0.74790241, \infty)$ in the first derivative $30 x^{5} - 12 x^{3} + 2$ to check if the result is negative or positive.

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Step 6.1

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

$f' (0) = 30 {(0)}^{5} - 12 {(0)}^{3} + 2$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$f' (0) = 30 \cdot 0 - 12 {(0)}^{3} + 2$

Step 6.2.1.2

Multiply $30$ by $0$ .

$f' (0) = 0 - 12 {(0)}^{3} + 2$

Step 6.2.1.3

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

$f' (0) = 0 - 12 \cdot 0 + 2$

Step 6.2.1.4

Multiply $- 12$ by $0$ .

$f' (0) = 0 + 0 + 2$

Step 6.2.2

Simplify by adding numbers.

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Step 6.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 2$

Step 6.2.2.2

Add $0$ and $2$ .

$f' (0) = 2$

Step 6.2.3

The final answer is $2$ .

$2$

Step 7

Since the first derivative changed signs from negative to positive around $x = - 0.74790241$ , then there is a turning point at $x = - 0.74790241$ .

Step 8

Find the y-coordinate of $- 0.74790241$ to find the turning point.

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Step 8.1

Find $f (- 0.74790241)$ to find the y-coordinate of $- 0.74790241$ .

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Step 8.1.1

Replace the variable $x$ with $- 0.74790241$ in the expression.

$f (- 0.74790241) = 5 {(- 0.74790241)}^{6} - 3 {(- 0.74790241)}^{4} + 2 (- 0.74790241) - 9$

Step 8.1.2

Simplify $5 {(- 0.74790241)}^{6} - 3 {(- 0.74790241)}^{4} + 2 (- 0.74790241) - 9$ .

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Step 8.1.2.1

Remove parentheses.

$5 {(- 0.74790241)}^{6} - 3 {(- 0.74790241)}^{4} + 2 (- 0.74790241) - 9$

Step 8.1.2.2

Simplify each term.

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Step 8.1.2.2.1

Raise $- 0.74790241$ to the power of $6$ .

$5 \cdot 0.17501272 - 3 {(- 0.74790241)}^{4} + 2 (- 0.74790241) - 9$

Step 8.1.2.2.2

Multiply $5$ by $0.17501272$ .

$0.8750636 - 3 {(- 0.74790241)}^{4} + 2 (- 0.74790241) - 9$

Step 8.1.2.2.3

Raise $- 0.74790241$ to the power of $4$ .

$0.8750636 - 3 \cdot 0.31288139 + 2 (- 0.74790241) - 9$

Step 8.1.2.2.4

Multiply $- 3$ by $0.31288139$ .

$0.8750636 - 0.93864419 + 2 (- 0.74790241) - 9$

Step 8.1.2.2.5

Multiply $2$ by $- 0.74790241$ .

$0.8750636 - 0.93864419 - 1.49580483 - 9$

Step 8.1.2.3

Simplify by subtracting numbers.

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Step 8.1.2.3.1

Subtract $0.93864419$ from $0.8750636$ .

$- 0.06358059 - 1.49580483 - 9$

Step 8.1.2.3.2

Subtract $1.49580483$ from $- 0.06358059$ .

$- 1.55938542 - 9$

Step 8.1.2.3.3

Subtract $9$ from $- 1.55938542$ .

$- 10.55938542$

Step 8.2

Write the $x$ and $y$ coordinates in point form.

$(- 0.74790241, - 10.55938542)$

Step 9