- #1
How do i move this over to homeworkBvU said:I wish you wouldn't write ##\ln ## as ##\operatorname {in}##
Please post in homework and tell us which relevant equations you think you will need to use ...
Considering he's trying to calculate derivatives using the basic definition, I think L'Hopital's Rule isn't fair game.Mark44 said:To complete the problem, I don't believe expanding the expression ##(x+h)^3−5## is any help. A better way to do this is to use L'Hopital's Rule.
Yes the basic definition is what i am after.vela said:Considering he's trying to calculate derivatives using the basic definition, I think L'Hopital's Rule isn't fair game.
After expanding ##(x+h)^3##, you can rewrite the argument of the log in the form (1+something). Go from there.
Vela here is what i have so farMichael Santos said:Yes the basic definition is what i am after.
Vela, will you be providing me with the rest of the answer?
Mark44 said:No, that's against forum rules. @vela has given you a strong hint that involves polynomial division.
Mark here is what i have so farMichael Santos said:Yes the basic definition is what i am after.
Vela, will you be providing me with the rest of the answer?
No i do notBvU said:Do you know a limit for $$\ln (1+\varepsilon)\over \varepsilon $$ when ##\varepsilon\downarrow 0 ## ?
Well i know it now, what can i do with it?BvU said:In that case things become difficult. What equations do you have in your toolkit that may help in this situation ? In the template they are called 'relevant equations'.
I suppose that refers to what you first said you did not knowMichael Santos said:Well i know it now,
? Well, what do you get if you apply your new knowledge to your last expression in post #9 ?Michael Santos said:No i do not
Michael Santos said:Mark here is what i have so far
$$\lim_{h\to 0} {\frac {\ln((x+h)^3 -5) - ln (x^3 - 5)} h} $$
What now?
$$\lim_{h\to 0}{{\frac {1} h} ×ln(1+{\frac {3x^2h+3xh^2+h^3} {x^3-5})}}$$
How doez that turn into the final answer ofRay Vickson said:You have
$$\frac{1}{h} \ln (1 + \underbrace{\frac{3 x^2 h + 3 x h^2 + h^3}{x^3-5}}_{H} ) $$
Here, ##H \to 0## whenever ##h \to 0##, so you have ##\ln(1+H)## with small ##|H|.##
You have $$\lim_{h\downarrow 0}\ {\ln (1+h )\over h} = 1$$ so what is $$\lim_{h\downarrow 0} \ {\ln (1+ \left(3x^2 \over x^3-5\right ) h )\over h} \quad \rm ? $$Michael Santos said:Well i know it now,
Michael Santos said:How doez that turn into the final answer of
$$\frac {3x^2} { \ x^3-5 \ }$$
SoRay Vickson said:Plug it in and work it out for yourself. Remember that you want to take the limit as ##h \to 0,## so lots of complicating terms will go away.
-1.39...BvU said:You have $$\lim_{h\downarrow 0}\ {\ln (1+h )\over h} = 1$$ so what is $$\lim_{h\downarrow 0} \ {\ln (1+ \left(3x^2 \over x^3-5\right ) h )\over h} \quad \rm ? $$
The difference quotient of ln (x^3 - 5) is the rate of change of the natural logarithm function at a specific value of x. It is used to calculate the slope of the tangent line at that point.
The difference quotient is calculated by taking the limit of the change in the function's output over the change in its input, as the change in the input approaches zero. In other words, it is the derivative of the function at a specific value of x.
The difference quotient is important because it is a fundamental concept in calculus and is used to calculate the slope of a curve at any given point. It is also used to find the instantaneous rate of change of a function, which has many real-world applications in fields such as physics and economics.
The difference quotient is directly related to the derivative. In fact, the derivative is defined as the limit of the difference quotient as the change in the input approaches zero. In other words, the derivative is the general formula for calculating the difference quotient at any given point.
Yes, the difference quotient can be used for any type of function, as long as the function is continuous and differentiable at the specific value of x. However, for some functions, the derivative may not exist at certain points, making it impossible to calculate the difference quotient at those points.