Calculus Examples

$y = 7 x \sin (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (7 x \sin (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x \sin (x)$ with respect to $x$ is $7 \frac{d}{d x} [x \sin (x)]$ .

$7 \frac{d}{d x} [x \sin (x)]$

Step 3.2

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

$7 (x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x])$

Step 3.3

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

$7 (x \cos (x) + \sin (x) \frac{d}{d x} [x])$

Step 3.4

Differentiate using the Power Rule.

Tap for more steps...

Step 3.4.1

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

$7 (x \cos (x) + \sin (x) \cdot 1)$

Step 3.4.2

Multiply $\sin (x)$ by $1$ .

$7 (x \cos (x) + \sin (x))$

Step 3.5

Apply the distributive property.

$7 x \cos (x) + 7 \sin (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 7 x \cos (x) + 7 \sin (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 7 x \cos (x) + 7 \sin (x)$