How do you find the derivative of x^(5/3) * ln(3x)x53ln(3x)?

1 Answer
May 16, 2016

x^(2/3)(1+5/3ln(3x))x23(1+53ln(3x))

Explanation:

differentiate using the color(blue)"product rule"product rule

If f(x) = g(x).h(x) then f'(x) = g(x)h'(x) + h(x)g'(x)...(A)
"-----------------------------------------------------"-----------------------------------------------------

g(x)=x^(5/3)rArrg'(x)=5/3x^(2/3)-"using the"color(blue)" power rule"
h(x)=ln(3x)rArrh'(x)=1/(3x)xx3=1/x

using d/dx(ln(f(x)))=1/(f(x))xxd/dxf(x)
"---------------------------------------------------------"
Substitute these values into (A)

f'(x)=x^(5/3)xx1/x+ln(3x)xx5/3x^(2/3)

=x^(5/3-1)+5/3x^(2/3)ln(3x)=x^(2/3)+5/3x^(2/3)ln(3x)

Taking out a common factor of x^(2/3)

rArrd/dx(x^(5/3).ln(3x))=x^(2/3)(1+5/3ln(3x))