Calculus Examples

y = 6 e^{x} cos (x)

$y = 6 e^{x} \cos (x)$ ,

(0, 6)

$(0, 6)$

Step 1

Find the first derivative and evaluate at

x = 0

$x = 0$ and

y = 6

$y = 6$ to find the slope of the tangent line.

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Step 1.1

Since

6

$6$ is constant with respect to

x

$x$ , the derivative of

6 e^{x} cos (x)

$6 e^{x} \cos (x)$ with respect to

x

$x$ is

6 \frac{d}{d x} [e^{x} cos (x)]

$6 \frac{d}{d x} [e^{x} \cos (x)]$ .

6 \frac{d}{d x} [e^{x} cos (x)]

$6 \frac{d}{d x} [e^{x} \cos (x)]$

Step 1.2

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = e^{x}

$f (x) = e^{x}$ and

g (x) = cos (x)

$g (x) = \cos (x)$ .

6 (e^{x} \frac{d}{d x} [cos (x)] + cos (x) \frac{d}{d x} [e^{x}])

$6 (e^{x} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [e^{x}])$

Step 1.3

The derivative of

cos (x)

$\cos (x)$ with respect to

x

$x$ is

- sin (x)

$- \sin (x)$ .

6 (e^{x} (- sin (x)) + cos (x) \frac{d}{d x} [e^{x}])

$6 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}])$

Step 1.4

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

$\frac{d}{d x} [a^{x}]$ is

a^{x} ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

6 (e^{x} (- sin (x)) + cos (x) e^{x})

$6 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

6 (e^{x} (- sin (x))) + 6 (cos (x) e^{x})

$6 (e^{x} (- \sin (x))) + 6 (\cos (x) e^{x})$

Step 1.5.2

Multiply

- 1

$- 1$ by

6

$6$ .

- 6 e^{x} sin (x) + 6 cos (x) e^{x}

$- 6 e^{x} \sin (x) + 6 \cos (x) e^{x}$

Step 1.5.3

Reorder terms.

- 6 e^{x} sin (x) + 6 e^{x} cos (x)

$- 6 e^{x} \sin (x) + 6 e^{x} \cos (x)$

- 6 e^{x} sin (x) + 6 e^{x} cos (x)

$- 6 e^{x} \sin (x) + 6 e^{x} \cos (x)$

Step 1.6

Evaluate the derivative at

x = 0

$x = 0$ .

- 6 e^{0} sin (0) + 6 e^{0} cos (0)

$- 6 e^{0} \sin (0) + 6 e^{0} \cos (0)$

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Anything raised to

0

$0$ is

1

$1$ .

- 6 \cdot 1 sin (0) + 6 e^{0} cos (0)

$- 6 \cdot 1 \sin (0) + 6 e^{0} \cos (0)$

Step 1.7.1.2

Multiply

$- 6$ by

$1$ .

$- 6 \sin (0) + 6 e^{0} \cos (0)$

Step 1.7.1.3

The exact value of

$\sin (0)$ is

$0$ .

$- 6 \cdot 0 + 6 e^{0} \cos (0)$

Step 1.7.1.4

Multiply

$- 6$ by

$0$ .

$0 + 6 e^{0} \cos (0)$

Step 1.7.1.5

Anything raised to

$0$ is

$1$ .

$0 + 6 \cdot 1 \cos (0)$

Step 1.7.1.6

Multiply

$6$ by

$1$ .

$0 + 6 \cos (0)$

Step 1.7.1.7

The exact value of

$\cos (0)$ is

$1$ .

$0 + 6 \cdot 1$

Step 1.7.1.8

Multiply

$6$ by

$1$ .

$0 + 6$

Step 1.7.2

Add

$0$ and

$6$ .

$6$

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

$6$ and a given point

$(0, 6)$ 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

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

$y - (6) = 6 \cdot (x - (0))$

Step 2.2

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

$y - 6 = 6 \cdot (x + 0)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Add

$x$ and

$0$ .

$y - 6 = 6 x$

Step 2.3.2

Add

$6$ to both sides of the equation.

$y = 6 x + 6$

Step 3