Question

How do I tell when and at how many points two functions have the same derivative?

Answer 1

the two functions are 3^x and x^3 if that helps.

Answer 2

Well, if you have two functions, f(x) and g(x), Normally, in general, \(\frac{d}{dx}f(x)\neq\frac{d}{dx}g(x)\) Unless \(g(x)=f(x)+C\) where \(C\) is a constant. Now of course, at specific points, the equation may hold true...but each derivative as a whole of f(x) and g(x) usually are not equal to each other. Ill demonstrate this with each of the function you gave me.

Answer 3

We have:
\[\left\{\eqalign{
&f(x)=3^x \\
&g(x)=x^3
}\right.\]
So we can get each of their derivatives:
\[\eqalign{
&f'(x)=\ln(3)3^x \\
&\\
&g'(x)=3x^2
}\]
So now we can see that:
\[\ln(3)3^x\neq3x^2\phantom{spce}(as\phantom{.}a\phantom{.}whole)\]
The only exception to this would be at a few x-values. These x-values will be where:
\[\ln(3)3^x-3x^2=0\]

Answer 4

Graphically, we can see that this is approximately at two points:

Now, we can't apparently EXACTLY isolate for the x-values. But we can approximate them by guessing values and checking to see if we get close to making the equation true. Make sense?