Solve Product Rule Problem: f(8)=7, g(8)=5, f'(8)=-4, g'(8)=6

[Rasine]

If f(8) = 7, g(8) = 5, f '(8) = -4, and g '(8) = 6, find the following numbers.

(a) Find (f + g)'(8).



i was tryig to apply the product rule to this by doing (f+g)(8)'+(f+g)'(8)...so 0+(f+g)'(8)...so that would be (-4+6)8...but that is not the right way to do it...

please help me get on the right path...

 

[murshid_islam]

shouldn't it be (f+g)'(8) = f'(8) + g'(8) = -4+6 = 2?

 

[mathman]

Rasine: Could you be looking for (fg)'(8)?

 

[HallsofIvy]

If f(8) = 7, g(8) = 5, f '(8) = -4, and g '(8) = 6, find the following numbers.

(a) Find (f + g)'(8).



i was tryig to apply the product rule to this by doing (f+g)(8)'+(f+g)'(8)...so 0+(f+g)'(8)...so that would be (-4+6)8...but that is not the right way to do it...

please help me get on the right path...

Why would you apply the product rule to a sum?

(f+ g)'(x)= f'(x)+ g'(x) so (f+ g)'(8)= -4+ 6= 2.

(fg)'(x)= f(x)g'(x)+ f'(x)g(x)= 7(6)+(-4)(5)= 42- 20= 22.

Where you under the impression that (f+ g)(8) meant f+ g times 8?? It doesn't, of course, it means the function f+ g applied to the number 8.

 

[arildno]

Your problem is quite simply stated:

You don't understand the NOTATION for functions.

When we write f(x), the WHOLE symbol "f(x)" denotes the function value, not just the "f"!
We reserve the single "f" symbol to denote the entire function, rather than any specific function value.

To ADD two functions, f and g, gives us a new function, conveniently symbolized as f+g, or equivalently, (f+g).(Note that this is "addition on a function space", where your elements are FUNCTIONS, rather than specific real numbers. Thus, "addition" has a different meaning, since the elements in a function space are not the same as numbers on the real number line, where "+" gained its first meaning. This overload of meaning given the symbol "+" is just something you have to learn&live with!)

The function value of the new function (f+g) is denoted as (f+g)(x).
Again, it is the ENTIRE symbol "(f+g)(x)" that denotes the function value, and not just a single part of it!

"Adding" two functions to form a new function should also include a recipe to compute the new function's function values in terms of the old functions' function values.
This recipe is given by the expression (f+g)(x)=f(x)+g(x).
That is, we gain (f+g)'s function values by adding together f's and g's function values.