Calculus Examples

$y = 2 x^{3} + 3 x^{2} - 12 x + 2$

Step 1

Set $y$ as a function of $x$ .

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

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{3} + 3 x^{2} - 12 x + 2$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

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

Step 2.2.2

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

$2 (3 x^{2}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$

Step 2.2.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$

Step 2.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$6 x^{2} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$

Step 2.3.2

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

$6 x^{2} + 3 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$

Step 2.3.3

Multiply $2$ by $3$ .

$6 x^{2} + 6 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$

Step 2.4

Evaluate $\frac{d}{d x} [- 12 x]$ .

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Step 2.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$6 x^{2} + 6 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.4.2

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

$6 x^{2} + 6 x - 12 \cdot 1 + \frac{d}{d x} [2]$

Step 2.4.3

Multiply $- 12$ by $1$ .

$6 x^{2} + 6 x - 12 + \frac{d}{d x} [2]$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$6 x^{2} + 6 x - 12 + 0$

Step 2.5.2

Add $6 x^{2} + 6 x - 12$ and $0$ .

$6 x^{2} + 6 x - 12$

Step 3

Set the derivative equal to $0$ then solve the equation $6 x^{2} + 6 x - 12 = 0$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Factor $6$ out of $6 x^{2} + 6 x - 12$ .

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Step 3.1.1.1

Factor $6$ out of $6 x^{2}$ .

$6 (x^{2}) + 6 x - 12 = 0$

Step 3.1.1.2

Factor $6$ out of $6 x$ .

$6 (x^{2}) + 6 (x) - 12 = 0$

Step 3.1.1.3

Factor $6$ out of $- 12$ .

$6 (x^{2}) + 6 x + 6 \cdot - 2 = 0$

Step 3.1.1.4

Factor $6$ out of $6 (x^{2}) + 6 x$ .

$6 (x^{2} + x) + 6 \cdot - 2 = 0$

Step 3.1.1.5

Factor $6$ out of $6 (x^{2} + x) + 6 \cdot - 2$ .

$6 (x^{2} + x - 2) = 0$

Step 3.1.2

Factor.

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Step 3.1.2.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 3.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.1.2.1.2

Write the factored form using these integers.

$6 ((x - 1) (x + 2)) = 0$

Step 3.1.2.2

Remove unnecessary parentheses.

$6 (x - 1) (x + 2) = 0$

Step 3.2

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

$x - 1 = 0$

$x + 2 = 0$

Step 3.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.4

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

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Step 3.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.5

The final solution is all the values that make $6 (x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

Step 4

Solve the original function $f (x) = 2 x^{3} + 3 x^{2} - 12 x + 2$ at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 {(1)}^{3} + 3 {(1)}^{2} - 12 \cdot 1 + 2$

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

One to any power is one.

$f (1) = 2 \cdot 1 + 3 {(1)}^{2} - 12 \cdot 1 + 2$

Step 4.2.1.2

Multiply $2$ by $1$ .

$f (1) = 2 + 3 {(1)}^{2} - 12 \cdot 1 + 2$

Step 4.2.1.3

One to any power is one.

$f (1) = 2 + 3 \cdot 1 - 12 \cdot 1 + 2$

Step 4.2.1.4

Multiply $3$ by $1$ .

$f (1) = 2 + 3 - 12 \cdot 1 + 2$

Step 4.2.1.5

Multiply $- 12$ by $1$ .

$f (1) = 2 + 3 - 12 + 2$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $2$ and $3$ .

$f (1) = 5 - 12 + 2$

Step 4.2.2.2

Subtract $12$ from $5$ .

$f (1) = - 7 + 2$

Step 4.2.2.3

Add $- 7$ and $2$ .

$f (1) = - 5$

Step 4.2.3

The final answer is $- 5$ .

$- 5$

Step 5

Solve the original function $f (x) = 2 x^{3} + 3 x^{2} - 12 x + 2$ at $x = - 2$ .

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Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

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

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 $- 2$ to the power of $3$ .

$f (- 2) = 2 \cdot - 8 + 3 {(- 2)}^{2} - 12 \cdot - 2 + 2$

Step 5.2.1.2

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 + 3 {(- 2)}^{2} - 12 \cdot - 2 + 2$

Step 5.2.1.3

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

$f (- 2) = - 16 + 3 \cdot 4 - 12 \cdot - 2 + 2$

Step 5.2.1.4

Multiply $3$ by $4$ .

$f (- 2) = - 16 + 12 - 12 \cdot - 2 + 2$

Step 5.2.1.5

Multiply $- 12$ by $- 2$ .

$f (- 2) = - 16 + 12 + 24 + 2$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 16$ and $12$ .

$f (- 2) = - 4 + 24 + 2$

Step 5.2.2.2

Add $- 4$ and $24$ .

$f (- 2) = 20 + 2$

Step 5.2.2.3

Add $20$ and $2$ .

$f (- 2) = 22$

Step 5.2.3

The final answer is $22$ .

$22$

Step 6

The horizontal tangent lines on function $f (x) = 2 x^{3} + 3 x^{2} - 12 x + 2$ are $y = - 5, y = 22$ .

$y = - 5, y = 22$

Step 7