Calculus Examples

$y = π \cos (6 x + 0.71)$

Step 1

Since $π$ is constant with respect to $x$ , the derivative of $π \cos (6 x + 0.71)$ with respect to $x$ is $π \frac{d}{d x} [\cos (6 x + 0.71)]$ .

$π \frac{d}{d x} [\cos (6 x + 0.71)]$

Step 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 $g (x) = 6 x + 0.71$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x + 0.71$ .

$π (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [6 x + 0.71])$

Step 2.2

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

$π (- \sin (u) \frac{d}{d x} [6 x + 0.71])$

Step 2.3

Replace all occurrences of $u$ with $6 x + 0.71$ .

$π (- \sin (6 x + 0.71) \frac{d}{d x} [6 x + 0.71])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $6 x + 0.71$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [0.71]$ .

$π (- \sin (6 x + 0.71) (\frac{d}{d x} [6 x] + \frac{d}{d x} [0.71]))$

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$π (- \sin (6 x + 0.71) (6 \frac{d}{d x} [x] + \frac{d}{d x} [0.71]))$

Step 3.3

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

$π (- \sin (6 x + 0.71) (6 \cdot 1 + \frac{d}{d x} [0.71]))$

Step 3.4

Multiply $6$ by $1$ .

$π (- \sin (6 x + 0.71) (6 + \frac{d}{d x} [0.71]))$

Step 3.5

Since $0.71$ is constant with respect to $x$ , the derivative of $0.71$ with respect to $x$ is $0$ .

$π (- \sin (6 x + 0.71) (6 + 0))$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $6$ and $0$ .

$π (- \sin (6 x + 0.71) \cdot 6)$

Step 3.6.2

Multiply $6$ by $- 1$ .

$π (- 6 \sin (6 x + 0.71))$

Step 3.6.3

Reorder the factors of $π (- 6 \sin (6 x + 0.71))$ .

$- 6 π \sin (6 x + 0.71)$