Use the following result to find the derivative of the function:

Question

d/dx[|u|]=u'u/|u|,u =/ 0.

h(x)=|6x|cos(8x)

I got h'(x)=-8|6x|sin8x+36cos(8x)/|6x| but i got marked wrong. Help ?

Answer 1

|U| = sqrt(U^2) = (U^2)^(1/2) <--- yes they are the exact same function

so the derivative is (1/2) (U^2) ^(-1/2)* 2U * dU <--- power rule and chain rule

= U * dU/ sqrt(U^2) <-- moves negative exponent to denominator

= U*dU / |U| = U U' / |U|, <-- substitutes du = U' and sqrt(U^2) = |U|

provided U is not zero

That's where the formula comes from....

Also note that |kX| = |k||x| = k|x| where k is a positive constant

Next,

f(x)= |6x|cos(8x) = S(x)*T(x) where S(x) = |6x| and T(x) = cos(8x)

Then

S'(x) = (6x) * 6 / |6x} = (36x)/|6x|

T'(x) = -8*sin(8x)

f'(x) = S * T' + S' * T <--- product rule

= |6x| * -8*sin(8x) + (36x)/|6x| * (cos(8x))

= 6 |x| * -8 sin(8x) + (36x)/ (6|x|) * cos(8x)

= -48|x| sin(8x) + cos(8x)*6x/|x|

= cos(8x)*6x/|x| - 48|x| sin(8x)

= [cos(8x)*6x - 48|x|^2 sin(8x)] / |x|