Calculus Examples

$y = {(1 + 5 x)}^{15}$ , $(0, 1)$

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 $f (x) = x^{15}$ and $g (x) = 1 + 5 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $1 + 5 x$ .

$\frac{d}{d u} [u^{15}] \frac{d}{d x} [1 + 5 x]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 15$ .

$15 u^{14} \frac{d}{d x} [1 + 5 x]$

Step 1.1.3

Replace all occurrences of $u$ with $1 + 5 x$ .

$15 {(1 + 5 x)}^{14} \frac{d}{d x} [1 + 5 x]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $1 + 5 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [5 x]$ .

$15 {(1 + 5 x)}^{14} (\frac{d}{d x} [1] + \frac{d}{d x} [5 x])$

Step 1.2.2

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

$15 {(1 + 5 x)}^{14} (0 + \frac{d}{d x} [5 x])$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [5 x]$ .

$15 {(1 + 5 x)}^{14} \frac{d}{d x} [5 x]$

Step 1.2.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$15 {(1 + 5 x)}^{14} (5 \frac{d}{d x} [x])$

Step 1.2.5

Multiply $5$ by $15$ .

$75 {(1 + 5 x)}^{14} \frac{d}{d x} [x]$

Step 1.2.6

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

$75 {(1 + 5 x)}^{14} \cdot 1$

Step 1.2.7

Multiply $75$ by $1$ .

$75 {(1 + 5 x)}^{14}$

Step 1.3

Evaluate the derivative at $x = 0$ .

$75 {(1 + 5 (0))}^{14}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $5$ by $0$ .

$75 {(1 + 0)}^{14}$

Step 1.4.2

Add $1$ and $0$ .

$75 \cdot 1^{14}$

Step 1.4.3

One to any power is one.

$75 \cdot 1$

Step 1.4.4

Multiply $75$ by $1$ .

$75$

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 $75$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 75 \cdot (x - (0))$

Step 2.2

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

$y - 1 = 75 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $x$ and $0$ .

$y - 1 = 75 x$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = 75 x + 1$

Step 3