Find the Derivative - d/dx x^(5sin(x))

$x^{5 \sin (x)}$

Step 1

Use the properties of logarithms to simplify the differentiation.

Rewrite $x^{5 \sin (x)}$ as $e^{\ln (x^{5 \sin (x)})}$ .

$\frac{d}{d x} [e^{\ln (x^{5 \sin (x)})}]$

Expand $\ln (x^{5 \sin (x)})$ by moving $5 \sin (x)$ outside the logarithm.

$\frac{d}{d x} [e^{5 \sin (x) \ln (x)}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 \sin (x) \ln (x)$ .

To apply the Chain Rule, set $u$ as $5 \sin (x) \ln (x)$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 \sin (x) \ln (x)]$

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [5 \sin (x) \ln (x)]$

Replace all occurrences of $u$ with $5 \sin (x) \ln (x)$ .

$e^{5 \sin (x) \ln (x)} \frac{d}{d x} [5 \sin (x) \ln (x)]$

Step 3

Differentiate using the Constant Multiple Rule.

Since $5$ is constant with respect to $x$ , the derivative of $5 \sin (x) \ln (x)$ with respect to $x$ is $5 \frac{d}{d x} [\sin (x) \ln (x)]$ .

$e^{5 \sin (x) \ln (x)} (5 \frac{d}{d x} [\sin (x) \ln (x)])$

Move $5$ to the left of $e^{5 \sin (x) \ln (x)}$ .

$5 \cdot e^{5 \sin (x) \ln (x)} \frac{d}{d x} [\sin (x) \ln (x)]$

Step 4

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 $f (x) = \sin (x)$ and $g (x) = \ln (x)$ .

$5 e^{5 \sin (x) \ln (x)} (\sin (x) \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [\sin (x)])$

Step 5

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$5 e^{5 \sin (x) \ln (x)} (\sin (x) \frac{1}{x} + \ln (x) \frac{d}{d x} [\sin (x)])$

Step 6

Combine $\sin (x)$ and $\frac{1}{x}$ .

$5 e^{5 \sin (x) \ln (x)} (\frac{\sin (x)}{x} + \ln (x) \frac{d}{d x} [\sin (x)])$

Step 7

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

$5 e^{5 \sin (x) \ln (x)} (\frac{\sin (x)}{x} + \ln (x) \cos (x))$

Step 8

Simplify.

Apply the distributive property.

$5 e^{5 \sin (x) \ln (x)} \frac{\sin (x)}{x} + 5 e^{5 \sin (x) \ln (x)} (\ln (x) \cos (x))$

Combine terms.

Combine $\frac{\sin (x)}{x}$ and $5$ .

$\frac{\sin (x) \cdot 5}{x} e^{5 \sin (x) \ln (x)} + 5 e^{5 \sin (x) \ln (x)} (\ln (x) \cos (x))$

Combine $\frac{\sin (x) \cdot 5}{x}$ and $e^{5 \sin (x) \ln (x)}$ .

$\frac{\sin (x) \cdot 5 e^{5 \sin (x) \ln (x)}}{x} + 5 e^{5 \sin (x) \ln (x)} (\ln (x) \cos (x))$

Move $5$ to the left of $\sin (x)$ .

$\frac{5 \sin (x) e^{5 \sin (x) \ln (x)}}{x} + 5 e^{5 \sin (x) \ln (x)} \ln (x) \cos (x)$

Reorder terms.

$5 e^{5 \sin (x) \ln (x)} \cos (x) \ln (x) + \frac{5 \sin (x) e^{5 \sin (x) \ln (x)}}{x}$

Find the Derivative - d/dx x^(5sin(x))

Related Questions

Support LCMGCF.com by disabling your adblocker.