Derivative of an Inverse Function

Question

Question 18: If $$f(x)=\ln(x+1)$$ (which is one-to-one) and $$g$$ is its inverse, find $$g'(0)$$.
 
| Step | Calculation Summary |
|------|---------------------|
| 1 | Since $g(x)$ is the inverse of $f(x)=\ln(x+1)$, we have $f(g(x))=x$. |
| 2 | Differentiate to obtain $g'(x)=\frac{1}{f'(g(x))}$. |
| 3 | With $f'(x)=\frac{1}{x+1}$, then $f'(g(0))=\frac{1}{g(0)+1}$. |
| 4 | Since $f(g(0))=0$, then $g(0)=0$ and $f'(0)=1$, yielding $g'(0)=1$. |

Accepted Answer

$$1$$

Answer

$$\frac{1}{2}$$

Answer

$$0$$

Answer

$$-1$$

Step	Calculation Summary
1	Since \(g(x)\) is the inverse of \(f(x)=\ln(x+1)\), we have \(f(g(x))=x\).
2	Differentiate to obtain \(g'(x)=\frac{1}{f'(g(x))}\).
3	With \(f'(x)=\frac{1}{x+1}\), then \(f'(g(0))=\frac{1}{g(0)+1}\).
4	Since \(f(g(0))=0\), then \(g(0)=0\) and \(f'(0)=1\), yielding \(g'(0)=1\).

Rank	User	Correct Count	Attempt Count	Time	Score
#1	haneesh19.12	1	1	0m 00s	100
#2	kamalkaren31	1	1	2m 48s	-68
#3	trinacamarillo2008	1	1	3m 29s	-109

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