Calculus Examples

$y = 3 x - 2 \sqrt{x}$ , $(1, 1)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 2 \sqrt{x}]$

Step 1.2.3

Multiply $3$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$3 + \frac{d}{d x} [- 2 x^{\frac{1}{2}}]$

Step 1.3.2

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

$3 - 2 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 1.3.3

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

$3 - 2 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 1.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 - 2 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 1.3.5

Combine $- 1$ and $\frac{2}{2}$ .

$3 - 2 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.3.6

Combine the numerators over the common denominator.

$3 - 2 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 1.3.7

Simplify the numerator.

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Step 1.3.7.1

Multiply $- 1$ by $2$ .

$3 - 2 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 1.3.7.2

Subtract $2$ from $1$ .

$3 - 2 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 1.3.8

Move the negative in front of the fraction.

$3 - 2 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 1.3.9

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$3 - 2 \frac{x^{- \frac{1}{2}}}{2}$

Step 1.3.10

Combine $- 2$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$3 + \frac{- 2 x^{- \frac{1}{2}}}{2}$

Step 1.3.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 + \frac{- 2}{2 x^{\frac{1}{2}}}$

Step 1.3.12

Factor $2$ out of $- 2$ .

$3 + \frac{2 \cdot - 1}{2 x^{\frac{1}{2}}}$

Step 1.3.13

Cancel the common factors.

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Step 1.3.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$3 + \frac{2 \cdot - 1}{2 (x^{\frac{1}{2}})}$

Step 1.3.13.2

Cancel the common factor.

$3 + \frac{2 \cdot - 1}{2 x^{\frac{1}{2}}}$

Step 1.3.13.3

Rewrite the expression.

$3 + \frac{- 1}{x^{\frac{1}{2}}}$

Step 1.3.14

Move the negative in front of the fraction.

$3 - \frac{1}{x^{\frac{1}{2}}}$

Step 1.4

Evaluate the derivative at $x = 1$ .

$3 - \frac{1}{{(1)}^{\frac{1}{2}}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$3 - \frac{1}{1}$

Step 1.5.1.2

Cancel the common factor of $1$ .

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Step 1.5.1.2.1

Cancel the common factor.

$3 - \frac{1}{1}$

Step 1.5.1.2.2

Rewrite the expression.

$3 - 1 \cdot 1$

Step 1.5.1.3

Multiply $- 1$ by $1$ .

$3 - 1$

Step 1.5.2

Subtract $1$ from $3$ .

$2$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $2$ and a given point $(1, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 2 \cdot (x - (1))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 1 = 2 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $2 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + 2 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = 2 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = 2 x + 2 \cdot - 1$

Step 2.3.1.4

Multiply $2$ by $- 1$ .

$y - 1 = 2 x - 2$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $1$ to both sides of the equation.

$y = 2 x - 2 + 1$

Step 2.3.2.2

Add $- 2$ and $1$ .

$y = 2 x - 1$

Step 3