#y=f(x)#
#y=x^(2sin(x))#
The Power Rule for cannot be used here, because #x# is not raised to a constant power, rather it is raised to the power of another function. Whenever we encounter a function in the exponent, we should apply the natural logarithm to both sides :
#ln(y)=ln(x^(2sin(x)))#
Recall the exponent rule for logarithms, which states that #ln(x^a)=aln(x)#. This rule applies even if #a# is a function and not just a constant.
#ln(y)=2sin(x)ln(x)#
Differentiate both sides with respect to #x#. This means the chain rule will apply for differentiating #ln(y)#, giving us an instance of #dy/dx:#
#1/y * dy/dx=(2sin(x))/x+2ln(x)cos(x)#
Solve for #dy/dx# by multiplying both sides by #y#:
#dy/dx=y((2sin(x))/x+2cos(x)ln(x))#
Recall that #y=x^(2sin(x))#:
#dy/dx=x^(2sin(x))((2sin(x))/x+2cos(x)ln(x))#
#f'(x)=x^(2sin(x))((2sin(x))/x+2cos(x)ln(x))#
