Calculus Examples

$y = 6 x^{7} \cos (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (6 x^{7} \cos (x))$

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 $6$ is constant with respect to $x$ , the derivative of $6 x^{7} \cos (x)$ with respect to $x$ is $6 \frac{d}{d x} [x^{7} \cos (x)]$ .

$6 \frac{d}{d x} [x^{7} \cos (x)]$

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 $f (x) = x^{7}$ and $g (x) = \cos (x)$ .

$6 (x^{7} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{7}])$

Step 3.3

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

$6 (x^{7} (- \sin (x)) + \cos (x) \frac{d}{d x} [x^{7}])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$6 (x^{7} (- \sin (x)) + \cos (x) (7 x^{6}))$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Apply the distributive property.

$6 (x^{7} (- \sin (x))) + 6 (\cos (x) (7 x^{6}))$

Step 3.5.2

Combine terms.

Tap for more steps...

Step 3.5.2.1

Multiply $- 1$ by $6$ .

$- 6 (x^{7} (\sin (x))) + 6 (\cos (x) (7 x^{6}))$

Step 3.5.2.2

Multiply $7$ by $6$ .

$- 6 x^{7} \sin (x) + 42 \cos (x) x^{6}$

Step 3.5.3

Reorder terms.

$- 6 x^{7} \sin (x) + 42 x^{6} \cos (x)$

Step 4

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

$y' = - 6 x^{7} \sin (x) + 42 x^{6} \cos (x)$

Step 5

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

$\frac{d y}{d x} = - 6 x^{7} \sin (x) + 42 x^{6} \cos (x)$