Calculus Examples

$y = 3 x^{2} - 6 x$

Step 1

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

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

Step 2

Find the derivative.

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Step 2.1

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

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

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $3$ .

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

Step 2.3

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

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Step 2.3.1

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

$6 x - 6 \frac{d}{d x} [x]$

Step 2.3.2

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

$6 x - 6 \cdot 1$

Step 2.3.3

Multiply $- 6$ by $1$ .

$6 x - 6$

Step 3

Set the derivative equal to $0$ then solve the equation $6 x - 6 = 0$ .

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Step 3.1

Add $6$ to both sides of the equation.

$6 x = 6$

Step 3.2

Divide each term in $6 x = 6$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 x = 6$ by $6$ .

$\frac{6 x}{6} = \frac{6}{6}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{6}{6}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{6}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $6$ by $6$ .

$x = 1$

Step 4

Solve the original function $f (x) = 3 x^{2} - 6 x$ at $x = 1$ .

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Step 4.1

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

$f (1) = 3 {(1)}^{2} - 6 \cdot 1$

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.

$f (1) = 3 \cdot 1 - 6 \cdot 1$

Step 4.2.1.2

Multiply $3$ by $1$ .

$f (1) = 3 - 6 \cdot 1$

Step 4.2.1.3

Multiply $- 6$ by $1$ .

$f (1) = 3 - 6$

Step 4.2.2

Subtract $6$ from $3$ .

$f (1) = - 3$

Step 4.2.3

The final answer is $- 3$ .

$- 3$

Step 5

The horizontal tangent line on function $f (x) = 3 x^{2} - 6 x$ is $y = - 3$ .

$y = - 3$

Step 6