# Function f(x)Find the derivative of the following function:f (x)= x^8 sin 5x

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It is given that f(x) = x^8 sin 5x, we need to find f'(x). We can use the product rule.

f'(x) = (x^8)' sin 5x + x^8(sin 5x)'

=> 8*x^7*(sin 5x) + 5x^8(cos 5x)

**The required derivative is 8*x^7(sin 5x) + 5x^8(cos 5x)**

To calculate the first derivative of the given function, we'll use the product rule and the chain rule:

f'(x) = (x^8)*(sin 5x)

We'll have 2 functions g and h:

(g*h)' = g'*h + g*h'

We'll put g = x^8 => g' = 8x^7

We'll put h = sin 5x => h' = (cos 5x)*(5x)'

h' = 5cos 5x

We'll substitute g,h,g',h' in the expression of (g*h)':

(g*h)' = 8x^7*(sin 5x) + x^8*(5cos 5x)

We'll factorize by x^7:

f'(x) = x^7[8(sin 5x) + x*(5cos 5x)]