Calculus Examples

$y = {(3 x - 1)}^{- 6}$ , $(0, 1)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 6}$ and $g (x) = 3 x - 1$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

$\frac{d}{d u} [u^{- 6}] \frac{d}{d x} [3 x - 1]$

Step 1.1.2

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

$- 6 u^{- 7} \frac{d}{d x} [3 x - 1]$

Step 1.1.3

Replace all occurrences of $u$ with $3 x - 1$ .

$- 6 {(3 x - 1)}^{- 7} \frac{d}{d x} [3 x - 1]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$- 6 {(3 x - 1)}^{- 7} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])$

Step 1.2.2

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

$- 6 {(3 x - 1)}^{- 7} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 1.2.3

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

$- 6 {(3 x - 1)}^{- 7} (3 \cdot 1 + \frac{d}{d x} [- 1])$

Step 1.2.4

Multiply $3$ by $1$ .

$- 6 {(3 x - 1)}^{- 7} (3 + \frac{d}{d x} [- 1])$

Step 1.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 6 {(3 x - 1)}^{- 7} (3 + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $3$ and $0$ .

$- 6 {(3 x - 1)}^{- 7} \cdot 3$

Step 1.2.6.2

Multiply $3$ by $- 6$ .

$- 18 {(3 x - 1)}^{- 7}$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 18 \frac{1}{{(3 x - 1)}^{7}}$

Step 1.3.2

Combine $- 18$ and $\frac{1}{{(3 x - 1)}^{7}}$ .

$\frac{- 18}{{(3 x - 1)}^{7}}$

Step 1.3.3

Move the negative in front of the fraction.

$- \frac{18}{{(3 x - 1)}^{7}}$

Step 1.4

Evaluate the derivative at $x = 0$ .

$- \frac{18}{{(3 (0) - 1)}^{7}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Multiply $3$ by $0$ .

$- \frac{18}{{(0 - 1)}^{7}}$

Step 1.5.1.2

Subtract $1$ from $0$ .

$- \frac{18}{{(- 1)}^{7}}$

Step 1.5.1.3

Raise $- 1$ to the power of $7$ .

$- \frac{18}{- 1}$

Step 1.5.2

Simplify the expression.

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Step 1.5.2.1

Divide $18$ by $- 1$ .

$- - 18$

Step 1.5.2.2

Multiply $- 1$ by $- 18$ .

$18$

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 $18$ and a given point $(0, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 18 \cdot (x - (0))$

Step 2.2

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

$y - 1 = 18 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $x$ and $0$ .

$y - 1 = 18 x$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = 18 x + 1$

Step 3