Calculus Examples

$y = \sin (7 x) + \sin^{2} (7 x)$ , $(0, 0)$

Step 1

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

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Step 1.1

By the Sum Rule, the derivative of $\sin (7 x) + \sin^{2} (7 x)$ with respect to $x$ is $\frac{d}{d x} [\sin (7 x)] + \frac{d}{d x} [\sin^{2} (7 x)]$ .

$\frac{d}{d x} [\sin (7 x)] + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2

Evaluate $\frac{d}{d x} [\sin (7 x)]$ .

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Step 1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 7 x$ .

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Step 1.2.1.1

To apply the Chain Rule, set $u_{1}$ as $7 x$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [7 x] + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [7 x] + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2.1.3

Replace all occurrences of $u_{1}$ with $7 x$ .

$\cos (7 x) \frac{d}{d x} [7 x] + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\cos (7 x) (7 \frac{d}{d x} [x]) + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2.3

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

$\cos (7 x) (7 \cdot 1) + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2.4

Multiply $7$ by $1$ .

$\cos (7 x) \cdot 7 + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.2.5

Move $7$ to the left of $\cos (7 x)$ .

$7 \cos (7 x) + \frac{d}{d x} [\sin^{2} (7 x)]$

Step 1.3

Evaluate $\frac{d}{d x} [\sin^{2} (7 x)]$ .

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Step 1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (7 x)$ .

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Step 1.3.1.1

To apply the Chain Rule, set $u_{2}$ as $\sin (7 x)$ .

$7 \cos (7 x) + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sin (7 x)]$

Step 1.3.1.2

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

$7 \cos (7 x) + 2 u_{2} \frac{d}{d x} [\sin (7 x)]$

Step 1.3.1.3

Replace all occurrences of $u_{2}$ with $\sin (7 x)$ .

$7 \cos (7 x) + 2 \sin (7 x) \frac{d}{d x} [\sin (7 x)]$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 7 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u_{3}$ as $7 x$ .

$7 \cos (7 x) + 2 \sin (7 x) (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [7 x])$

Step 1.3.2.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$7 \cos (7 x) + 2 \sin (7 x) (\cos (u_{3}) \frac{d}{d x} [7 x])$

Step 1.3.2.3

Replace all occurrences of $u_{3}$ with $7 x$ .

$7 \cos (7 x) + 2 \sin (7 x) (\cos (7 x) \frac{d}{d x} [7 x])$

Step 1.3.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \cos (7 x) + 2 \sin (7 x) (\cos (7 x) (7 \frac{d}{d x} [x]))$

Step 1.3.4

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

$7 \cos (7 x) + 2 \sin (7 x) (\cos (7 x) (7 \cdot 1))$

Step 1.3.5

Multiply $7$ by $1$ .

$7 \cos (7 x) + 2 \sin (7 x) (\cos (7 x) \cdot 7)$

Step 1.3.6

Move $7$ to the left of $\cos (7 x)$ .

$7 \cos (7 x) + 2 \sin (7 x) (7 \cdot \cos (7 x))$

Step 1.3.7

Multiply $7$ by $2$ .

$7 \cos (7 x) + 14 \sin (7 x) \cos (7 x)$

Step 1.4

Reorder terms.

$14 \cos (7 x) \sin (7 x) + 7 \cos (7 x)$

Step 1.5

Evaluate the derivative at $x = 0$ .

$14 \cos (7 (0)) \sin (7 (0)) + 7 \cos (7 (0))$

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $7$ by $0$ .

$14 \cos (0) \sin (7 (0)) + 7 \cos (7 (0))$

Step 1.6.1.2

The exact value of $\cos (0)$ is $1$ .

$14 \cdot 1 \sin (7 (0)) + 7 \cos (7 (0))$

Step 1.6.1.3

Multiply $14$ by $1$ .

$14 \sin (7 (0)) + 7 \cos (7 (0))$

Step 1.6.1.4

Multiply $7$ by $0$ .

$14 \sin (0) + 7 \cos (7 (0))$

Step 1.6.1.5

The exact value of $\sin (0)$ is $0$ .

$14 \cdot 0 + 7 \cos (7 (0))$

Step 1.6.1.6

Multiply $14$ by $0$ .

$0 + 7 \cos (7 (0))$

Step 1.6.1.7

Multiply $7$ by $0$ .

$0 + 7 \cos (0)$

Step 1.6.1.8

The exact value of $\cos (0)$ is $1$ .

$0 + 7 \cdot 1$

Step 1.6.1.9

Multiply $7$ by $1$ .

$0 + 7$

Step 1.6.2

Add $0$ and $7$ .

$7$

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 $7$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 7 \cdot (x - (0))$

Step 2.2

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

$y + 0 = 7 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 7 \cdot (x + 0)$

Step 2.3.2

Add $x$ and $0$ .

$y = 7 x$

Step 3