Composite exponential function differentiation

Calculus Tutorial

Composite exponential function differentiation

Intro

When we differentiate the composite exponential function, we want to use the product rule and the chain rule.

Sample Problem

y=(ln(x))x
find dy/dx

Solution

y=(ln(x))x
ln(y)=ln(ln(x))x take (ln) both side
ln(y)=x(ln(ln(x)) ln(a)m=m.ln(a)
d/dx(ln(y))=d/dx(x(ln(ln(x))) take derivative both side
1/y.dy/dx=x.d/dx(ln(ln(x)))+ln(ln(x)).d/dx(x) product rule
1/y.dy/dx=x.1/ln(x).1/x+ln(ln(x)).1 chain rule
1/y.dy/dx=1/ln(x)+ln(ln(x)) simplify
dy/dx=y.(1/ln(x)+ln(ln(x))) multiply by y both side
dy/dx=ln(x)x.(1/ln(x)+ln(ln(x))) substituting y



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