Calculus Examples

$x \cos (π x) + \sin (x)$

Step 1

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

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (π x)$ .

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

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 $g (x) = π x$ .

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $π$ by $1$ .

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

Step 2.7

Multiply $\cos (π x)$ by $1$ .

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

Step 3

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

$x (- \sin (π x) π) + \cos (π x) + \cos (x)$

Step 4

Reorder terms.

$- π x \sin (π x) + \cos (π x) + \cos (x)$