Derivatives?

Question

(Sin x Cos x)/(7x^5)?

Answer 1

The function f(x) = (sinx cosx)/(7x⁵)

Apply derivative with respect to x.

d/dx[f(x)] = d/dx[(sinx cosx)/(7x⁵)]

Recall the formulae :

Quotient rule : d/dx(u/v) = (vu'- uv')/v²

Product rule : d/dx(uv) = uv' + vu'

d/dx (xⁿ) = nx^{n- 1}

f'(x) = [(7x⁵)(sinx cosx)' - (sinx cosx)(7x⁵)']/(7x⁵)²

 = {(7x⁵)[sinx (-sinx)+ (cosx)(cosx)] - (sinx cosx)(35x⁴)}/(49x¹⁰)

= {7x⁵[- sin²x + cos²x] - 35x⁴sinx cosx}/(49x¹⁰)

{ Apply Pythagorean identities sin²x + cos²x = 1 }

= {7x⁵[- sin²x + 1 - sin²x] - 35x⁴sinx cosx}/(49x¹⁰)

= 7x⁴{x[1 - 2 sin²x] - 5 sinx cosx}/(49x¹⁰)

{Apply double angle formula cos(2x) = 1- 2sin²x}

f'(x) = [xcos(2x) - 5 sinx cosx]/(7x⁶)

{Apply double angle formula sin(2x) = 2sinxcosx}

f'(x) = [xcos(2x) - (5/2) sin2x]/(7x⁶)

Derivative of (sinx cosx)/(7x⁵) is [xcos(2x) - (5/2) sin2x]/(7x⁶).