Find the derivative below.

Question

Find the derivative below.

 

??=4 log_3 (??^2−1)

Answer 1

For a log with a base other than e, the derivative rule picks up one extra piece. The derivative of log_b(u) is u′/(u · ln b), so the base shows up as an ln b sitting down in the denominator.

Here our inside function is u = x² − 1, so u′ = 2x, and the 4 out front is just a constant that rides along:

dy/dx = 4 · 2x/((x² − 1) · ln 3)

which simplifies to:

dy/dx = 8x/((x² − 1) · ln 3)

Remember, ln 3 is a constant, not a function of x, so it never gets differentiated. It just sits in the denominator for the whole problem. A common slip here is to leave it out entirely, which is what happens when someone uses the natural log rule on a log that is not base e. If you want to see where that ln 3 comes from, you can rewrite log₃(u) as ln(u)/ln 3 using the change of base formula, and then it is just the ordinary ln derivative with a constant divided into it.

One more thing worth noticing is the domain. We need x² − 1 > 0 for the log to be defined, so x has to satisfy |x| > 1, which means our answer only applies on (−∞, −1) ∪ (1, ∞).

So, our final answer is dy/dx = 8x/((x² − 1) · ln 3).