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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)]
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