Calculus Examples

$y = 7 + 5 \cos (0.503 (x - 6.75))$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $7 + 5 \cos (0.503 (x - 6.75))$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [5 \cos (0.503 (x - 6.75))]$ .

$\frac{d}{d x} [7] + \frac{d}{d x} [5 \cos (0.503 (x - 6.75))]$

Step 1.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [5 \cos (0.503 (x - 6.75))]$

Step 2

Evaluate $\frac{d}{d x} [5 \cos (0.503 (x - 6.75))]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (0.503 (x - 6.75))$ with respect to $x$ is $5 \frac{d}{d x} [\cos (0.503 (x - 6.75))]$ .

$0 + 5 \frac{d}{d x} [\cos (0.503 (x - 6.75))]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 0.503 (x - 6.75)$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $0.503 (x - 6.75)$ .

$0 + 5 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [0.503 (x - 6.75)])$

Step 2.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$0 + 5 (- \sin (u) \frac{d}{d x} [0.503 (x - 6.75)])$

Step 2.2.3

Replace all occurrences of $u$ with $0.503 (x - 6.75)$ .

$0 + 5 (- \sin (0.503 (x - 6.75)) \frac{d}{d x} [0.503 (x - 6.75)])$

Step 2.3

Since $0.503$ is constant with respect to $x$ , the derivative of $0.503 (x - 6.75)$ with respect to $x$ is $0.503 \frac{d}{d x} [x - 6.75]$ .

$0 + 5 (- \sin (0.503 (x - 6.75)) (0.503 \frac{d}{d x} [x - 6.75]))$

Step 2.4

By the Sum Rule, the derivative of $x - 6.75$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6.75]$ .

$0 + 5 (- \sin (0.503 (x - 6.75)) (0.503 (\frac{d}{d x} [x] + \frac{d}{d x} [- 6.75])))$

Step 2.5

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

$0 + 5 (- \sin (0.503 (x - 6.75)) (0.503 (1 + \frac{d}{d x} [- 6.75])))$

Step 2.6

Since $- 6.75$ is constant with respect to $x$ , the derivative of $- 6.75$ with respect to $x$ is $0$ .

$0 + 5 (- \sin (0.503 (x - 6.75)) (0.503 (1 + 0)))$

Step 2.7

Add $1$ and $0$ .

$0 + 5 (- \sin (0.503 (x - 6.75)) (0.503 \cdot 1))$

Step 2.8

Multiply $0.503$ by $1$ .

$0 + 5 (- \sin (0.503 (x - 6.75)) \cdot 0.503)$

Step 2.9

Multiply $0.503$ by $- 1$ .

$0 + 5 (- 0.503 \sin (0.503 (x - 6.75)))$

Step 2.10

Multiply $- 0.503$ by $5$ .

$0 - 2.515 \sin (0.503 (x - 6.75))$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$0 - 2.515 \sin (0.503 x + 0.503 \cdot - 6.75)$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $0.503$ by $- 6.75$ .

$0 - 2.515 \sin (0.503 x - 3.39525)$

Step 3.2.2

Subtract $2.515 \sin (0.503 x - 3.39525)$ from $0$ .

$- 2.515 \sin (0.503 x - 3.39525)$