Logarithmic differentiation question

Question

I came across this problem in my textbook:

I am a bit confused with it though because I thought the rule for this type of problem was:

If the above rule is true, shouldn't the differentiation process look like this:

Where is that random $2x$ coming from in the numerator?

Answer 1

Hint: The chain rule. When we have composition of functions, such as $f(g(x))$, the derivative is $f^'(g(x))\cdot g^'(x)$. (We multiply by the derivative of the inner function). The derivative of the "inside" in your case is the derivative of the function $x^2-1$, which is $2x$.

Hope that helps,

Answer 2

Let $u = x^2-1 $. Then $y=\frac{1}{2} \log_5 u$.
So by the chain rule, we have

$$ \frac{\text{d}y}{\text{d}x} = \frac{\text{d}y}{\text{d}u}\cdot \frac{\text{d}u}{\text{d}x}$$

You can then find $\frac{\text{d}y}{\text{d}u}$ by the rule you gave. Then make the necessary substitutions.
Hope this helps.