Questions: Find the derivative of (f(x)). (f(x)=x^5 sin (x)) (f^prime(x)=)

Transcript text: Find the derivative of $f(x)$. \[ \begin{array}{l} f(x)=x^{5} \sin (x) \\ f^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

To find the derivative of the function $ f(x) = x^5 \sin(x) $, we will use the product rule. The product rule states that if you have a function that is the product of two functions, $ u(x) $ and $ v(x) $, then the derivative is given by $ (uv)' = u'v + uv' $. Here, $ u(x) = x^5 $ and $ v(x) = \sin(x) $.

Solution Approach

Identify the two functions $ u(x) $ and $ v(x) $.
Compute the derivatives $ u'(x) $ and $ v'(x) $.
Apply the product rule formula.

Step 1: Identify the Functions

We have the function $ f(x) = x^5 \sin(x) $. Here, we identify:

$ u(x) = x^5 $
$ v(x) = \sin(x) $

Step 2: Compute the Derivatives

Next, we compute the derivatives of $ u(x) $ and $ v(x) $:

$ u'(x) = 5x^4 $
$ v'(x) = \cos(x) $

Step 3: Apply the Product Rule

Using the product rule, we find the derivative $ f'(x) $: \[ f'(x) = u'v + uv' = (5x^4)(\sin(x)) + (x^5)(\cos(x)) \] This simplifies to: \[ f'(x) = 5x^4 \sin(x) + x^5 \cos(x) \]