Algebra Examples

$y = x^{8 \cos (X)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d X} (y) = \frac{d}{d X} (x^{8 \cos (X)})$

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

Differentiate using the Generalized Power Rule which states that $\frac{d}{d X} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = x$ and $g = 8 \cos (X)$ .

$\frac{d}{d X} [x] (8 \cos (X) x^{8 \cos (X) - 1}) + \frac{d}{d X} [8 \cos (X)] (x^{8 \cos (X)} \ln (x))$

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

$0 (8 \cos (X) x^{8 \cos (X) - 1}) + \frac{d}{d X} [8 \cos (X)] x^{8 \cos (X)} \ln (x)$

Step 3.2.2

Simplify the expression.

Tap for more steps...

Step 3.2.2.1

Multiply $8$ by $0$ .

$0 (\cos (X) x^{8 \cos (X) - 1}) + \frac{d}{d X} [8 \cos (X)] x^{8 \cos (X)} \ln (x)$

Step 3.2.2.2

Multiply $\cos (X)$ by $0$ .

$0 x^{8 \cos (X) - 1} + \frac{d}{d X} [8 \cos (X)] x^{8 \cos (X)} \ln (x)$

Step 3.2.2.3

Multiply $0$ by $x^{8 \cos (X) - 1}$ .

$0 + \frac{d}{d X} [8 \cos (X)] x^{8 \cos (X)} \ln (x)$

Step 3.2.2.4

Add $0$ and $\frac{d}{d X} [8 \cos (X)] x^{8 \cos (X)} \ln (x)$ .

$\frac{d}{d X} [8 \cos (X)] x^{8 \cos (X)} \ln (x)$

Step 3.2.3

Since $8$ is constant with respect to $X$ , the derivative of $8 \cos (X)$ with respect to $X$ is $8 \frac{d}{d X} [\cos (X)]$ .

$8 \frac{d}{d X} [\cos (X)] x^{8 \cos (X)} \ln (x)$

Step 3.3

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

$8 (- \sin (X)) x^{8 \cos (X)} \ln (x)$

Step 3.4

Simplify the expression.

Tap for more steps...

Step 3.4.1

Multiply $- 1$ by $8$ .

$- 8 \sin (X) x^{8 \cos (X)} \ln (x)$

Step 3.4.2

Reorder the factors of $- 8 \sin (X) x^{8 \cos (X)} \ln (x)$ .

$- 8 x^{8 \cos (X)} \sin (X) \ln (x)$

Step 4

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

$y' = - 8 x^{8 \cos (X)} \sin (X) \ln (x)$

Step 5

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

$\frac{d y}{d X} = - 8 x^{8 \cos (X)} \sin (X) \ln (x)$