How do I find this derivative?

Question

I need to find the derivative of this $[f(x)]^{g(x)}$.What rule of differentiation do I use ?. I don't think the power rule is applicable in this case.

You can write this as $e^{g(x) \ln f(x)}$, and then differentiate, maybe? — Sarvesh Ravichandran Iyer, May 02 '17 at 00:14

score 4 · Accepted Answer · answered May 02 '17 at 00:20

4

Well, logarithms are useful. $$h(x)=f(x)^{g(x)}$$ $$\ln(h(x))=g(x)\ln(f(x))$$ Now we can use chain rule and product rule. $$h'(x)\frac{1}{h(x)}=\frac{g(x)f'(x)}{f(x)}+g'(x)\ln{f(x)}$$ $$h'(x)=f(x)^{g(x)}\bigg(\frac{g(x)f'(x)}{f(x)}+g'(x)\ln{f(x)}\bigg)$$

answered May 02 '17 at 00:20

Isaac Browne

8,076

I remember asking this exact question in my calculus class. Try doing it for logarithms now! It uses a similar technique. – Isaac Browne May 02 '17 at 00:21

score 2 · Answer 2 · answered May 02 '17 at 00:42

2

Let $y=f^g$. By the multi-variable chain rule, we have...

$$\begin{align}\frac{dy}{dx}&=\frac{\partial y}{\partial f}\frac{df}{dx}+\frac{\partial y}{\partial g}\frac{dg}{dx}\\&=gf^{g-1}f'+\ln(f)f^gg'\\&=f(x)^{g(x)}\left(\frac{g(x)f'(x)}{f(x)}+\ln(f(x))g'(x)\right)\end{align}$$

answered May 02 '17 at 00:42

Simply Beautiful Art

74,685

That's the best way to take a derivative, +1 – Jaideep Khare May 02 '17 at 01:29

How do I find this derivative?

2 Answers2