# 2.2.1 AP Calculus Exam .... derivative with ln

• MHB
• karush
In summary: Since f(x)= 7x-3+ ln(x) is a continuous function, it is invertible- and the inverse will also be continuous. And since the derivative of the inverse is the reciprocal of the derivative of the original function, yes, g'(3)= -1/2, "(A)".In summary, if $f(x)=7x-3+\ln(x),$ then $f'(1)= -2, "(A)". karush Gold Member MHB If$f(x)=7x-3+\ln(x),$then$f'(1)=a.4\quad b. 5\quad c. 6\quad d. 7\quad e. 8$see if you can solve this before see the proposed solution since $$f'(x)=7+\dfrac{1}{x}$$ so then $$f'(x)=7+\dfrac{1}{1}=7+1=8\textit{ is (e)}$$ I "see"$7 + \dfrac{1}{x}\$ as the derivative ...

here's one for you ...

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I will go with karush's question instead since my brain is closer to that level of question.
$$\displaystyle f'(1)=7+\frac11=7+1=8$$

Monoxdifly said:
I will go with karush's question instead since my brain is closer to that level of question.
$$\displaystyle f'(1)=7+\frac11=7+1=8$$

D is 7

karush said:
D is 7

Oops, sorry. I mean E.

Duh, even though I went with the easy question I still ended up misread the options. Sometimes I think I have ADD.

Well you are probably
Better at math than I am

karush said:
Well you are probably
Better at math than I am

Nah. Otherwise, I would do skeeter's question instead.

For skeeter's question use the fact that $$\frac{df^{-1}}{dx}= \frac{1}{\frac{df}{dx}. We are asked to find g'(3) when$$g(x)= f^{-1}{x)[/tex]. We know that g(f(x))= x and see that f(6)= 3. Since f'(6)= -2, g'(3)= -1/2, "(A)".

Actually, the fact that the only derivative we are given f'(6)= -2 is pretty much a "give away"!

## 1. What is the derivative of ln(x)?

The derivative of ln(x) is 1/x.

## 2. How do you find the derivative of ln(u), where u is a function of x?

To find the derivative of ln(u), we use the chain rule and the fact that d/dx(ln(x)) = 1/x. The derivative is given by d/dx(ln(u)) = 1/u * du/dx.

## 3. Can the derivative of ln(x) be negative?

No, the derivative of ln(x) is always positive for x > 0. This is because the natural logarithm function is monotonically increasing, meaning it always increases as x increases.

## 4. What is the derivative of ln(x) at x = 1?

The derivative of ln(x) at x = 1 is 1. This can be seen by taking the limit as x approaches 1 of the derivative, which is equal to the limit as x approaches 1 of 1/x, which is equal to 1.

## 5. Can the derivative of ln(x) be undefined?

No, the derivative of ln(x) is defined for all x > 0. However, it is undefined at x = 0 as the natural logarithm function is not defined for x = 0.

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