Calculus Examples

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

Step 1

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

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

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $3$ .

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

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$3 x^{2} + 6 x + 0$

Step 2.3.2

Add $3 x^{2} + 6 x$ and $0$ .

$3 x^{2} + 6 x$

Step 3

Set the derivative equal to $0$ then solve the equation $3 x^{2} + 6 x = 0$ .

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Step 3.1

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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Step 3.1.1

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

$3 x (x) + 6 x = 0$

Step 3.1.2

Factor $3 x$ out of $6 x$ .

$3 x (x) + 3 x (2) = 0$

Step 3.1.3

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

$3 x (x + 2) = 0$

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 $0$ .

$x = 0$

$x + 2 = 0$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

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 $3 x (x + 2) = 0$ true.

$x = 0, - 2$

Step 4

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

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Step 4.1

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

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

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

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

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

Step 4.2.1.2

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

$f (0) = 0 + 3 \cdot 0 + 2$

Step 4.2.1.3

Multiply $3$ by $0$ .

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

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 2$

Step 4.2.2.2

Add $0$ and $2$ .

$f (0) = 2$

Step 4.2.3

The final answer is $2$ .

$2$

Step 5

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

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Step 5.1

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

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

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 $3$ .

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

Step 5.2.1.2

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

$f (- 2) = - 8 + 3 \cdot 4 + 2$

Step 5.2.1.3

Multiply $3$ by $4$ .

$f (- 2) = - 8 + 12 + 2$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 8$ and $12$ .

$f (- 2) = 4 + 2$

Step 5.2.2.2

Add $4$ and $2$ .

$f (- 2) = 6$

Step 5.2.3

The final answer is $6$ .

$6$

Step 6

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

$y = 2, y = 6$

Step 7