Calculus Examples

$π \cos (π x)$

Step 1

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

$π \frac{d}{d x} [\cos (π x)]$

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 $g (x) = π x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $π x$ .

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

Step 2.2

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

$π (- \sin (u) \frac{d}{d x} [π x])$

Step 2.3

Replace all occurrences of $u$ with $π x$ .

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

Step 3

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

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

Step 4

Raise $π$ to the power of $1$ .

$- \sin (π x) (π^{1} π \frac{d}{d x} [x])$

Step 5

Raise $π$ to the power of $1$ .

$- \sin (π x) (π^{1} π^{1} \frac{d}{d x} [x])$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7

Add $1$ and $1$ .

$- \sin (π x) (π^{2} \frac{d}{d x} [x])$

Step 8

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

$- \sin (π x) (π^{2} \cdot 1)$

Step 9

Simplify the expression.

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Step 9.1

Multiply $π^{2}$ by $1$ .

$- \sin (π x) π^{2}$

Step 9.2

Reorder the factors of $- \sin (π x) π^{2}$ .

$- π^{2} \sin (π x)$