Faster Higher-Order Derivatives

Question

so I'm taking a calc class and I got a question to calculate a derivative similar to this one: $\ln((x+1)^4(x+2)^7(x+8)^4)$.

I used chain rule and power rule to find that the answer was: $\frac{15x^2+115x+128}{(x+1)(x+2)(x+8)}$.

But it took me a really long time to do chain rule and then factor out the powers. I was wondering if, for this problem or just in general, there are faster methods and techniques to calculate higher order derivatives. I would really appreciate any help, and I'm pretty much just curious now too!

Answer 1

Use the product and power laws of logarithms. We have

\begin{align}\ln((x+1)^4(x+2)^7(x+8)^4)&=\ln(x+1)^4+\ln(x+2)^7+\ln(x+8)^4\\ &=4\ln(x+1)+7\ln(x+2)+4\ln(x+8) \end{align}

Therefore, its derivative is $$\frac4{x+1}+\frac7{x+2}+\frac4{x+8}.$$

Answer 2

The point of taking the logarithm in many cases is it splits up a product and kills powers. For example, $$\log((x^2+2)^2(x+3)^3)=\log((x^2+2)^2)+\log((x+3)^3)=2\log(x^2+2)+3\log(x+3)$$ It is easy to take the derivative of the right hand side of this.