Solving a Derivative Problem using Chain Rule and Logarithmic Differentiation

[S.R]

Homework Statement

Assume the notation log(a, x) implies log base a of x, where a is a constant (since I don't know LaTeX).

PROBLEM:
If y = [log(a, x^2)]^2, determine y'.

Homework Equations

Chain Rule and Logarithmic Differentiation

The Attempt at a Solution

y' = 2(log(a, x^2)) * (1/[(x^2)lna]) * (2x) = (8log(a,x))/(xlna)

Is this the correct approach and solution?

 

[gopher_p]

Homework Statement

Assume the notation log(a, x) implies log base a of x, where a is a constant (since I don't know LaTeX).

PROBLEM:
If y = [log(a, x^2)]^2, determine y'.

Homework Equations

Chain Rule and Logarithmic Differentiation

The Attempt at a Solution

y' = 2(log(a, x^2)) * (1/[(x^2)lna]) * (2x) = (8log(a,x))/(xlna)

Is this the correct approach and solution?

Your approach and answer are both correct. If I may offer an alternate approach, try using the log law $\log_ax^2=2\log_a x$ at the beginning and see how that changes the rest of the problem. It's my experience that making good use of log laws at the beginning of some calculus problems makes them a little more manageable.

 

[S.R]

Thanks for the response. I noticed the implementation of log laws in WolframAlpha's solution where log(a,x^2) was rewritten ln(x^2)/lna.

 

[rjr]

gopher_p is correct: using the log law log_a x² = 2log_a x does make for easier computation.

however, if you set u = log_a x² then your equation would become y = u²

so then, y'(x) = du/dx * 2u

the only "tricky" part is finding du/dx, but as S.R mentioned, you can just use the change of base formula for the logarithm...which makes it much easier to find du/dx...

u = log_a x²

= (log₁₀ x²)/(log₁₀ a)

= (log x²)/(log a)

...then just differentiate with respect to x to find du/dx

so yes, this is definitely the correct approach to the solution!

 

[HallsofIvy]

Of course, it is much simpler to first use the fact that $$log_a(x^2)= 2log_a(x)$$ and then differentiate.