# How do you find the derivative of #x^(log(base5)(x))#?

##### 1 Answer

#### Explanation:

#y=x^(log_5(x))#

Take the natural logarithm of both sides. (This is known as logarithmic differentiation.)

#ln(y)=ln(x^(log_5(x)))#

Use the rule:

#ln(y)=log_5(x)*ln(x)#

Now, rewrite

#ln(y)=ln(x)/ln(5)*ln(x)#

#ln(y)=(ln(x))^2/ln(5)#

Differentiate both sides. The chain rule will be needed on both sides of the equation.

#1/y*dy/dx=(2ln(x))/ln(5)*d/dxln(x)#

We already have used the derivative of

#1/x^(log_5(x))*dy/dx=(2ln(x))/(xln(5))#

Note that

#1/x^(log_5(x))*dy/dx=(2log_5(x))/x#

Multiplying both sides by

#dy/dx=(2x^(log_5(x))log_5(x))/x#

Note that

#dy/dx=2x^(log_5(x)-1)log_5(x)#