Find the Derivative - d/dx y = natural log of natural log of x^2

Problem

d()/d(x)*ln(ln(x2))

Solution

1. Identify the outer function and the inner functions to apply the chain rule. The expression is a composition of functions: ƒ*(g*(h(x))) where ƒ(u)=ln(u) g(v)=ln(v) and h(x)=x2

2. Apply the chain rule to the outermost natural log. The derivative of ln(u) is 1/u⋅d(u)/d(x)

d(ln(ln(x2)))/d(x)=1/ln(x2)⋅d(ln(x2))/d(x)

3. Apply the chain rule again to the inner natural log. The derivative of ln(x2) is 1/(x2)⋅d(x2)/d(x)

d(ln(ln(x2)))/d(x)=1/ln(x2)⋅1/(x2)⋅d(x2)/d(x)

4. Differentiate the innermost power function x2 using the power rule.

d(x2)/d(x)=2*x

5. Substitute the result back into the expression.

d(ln(ln(x2)))/d(x)=1/ln(x2)⋅1/(x2)⋅2*x

6. Simplify the expression by canceling x in the numerator and denominator.

d(ln(ln(x2)))/d(x)=(2*x)/(x2*ln(x2))

d(ln(ln(x2)))/d(x)=2/(x*ln(x2))

7. Apply log properties to further simplify if desired, noting that ln(x2)=2*ln(x)

d(ln(ln(x2)))/d(x)=2/(x⋅2*ln(x))

d(ln(ln(x2)))/d(x)=1/(x*ln(x))

Final Answer

d(ln(ln(x2)))/d(x)=1/(x*ln(x))

Want more problems? Check here!