How to prove the derivative is equal to it's original sum???

[Alexmcom]

f(x)=5x^3 and g(x)= 4x^2, prove that the derivative of the sum of the functions is equal to the sum of the derivative of each function:

. . . . .\(\displaystyle \left[\, f(x)\, +\, g(x)\, \right]'\, =\, f'(x)\, +\, g'(x)\)

So I don't know how to do this. I start off with getting the derivative of fx and gx so fx is 10x^2 and 8x and then I am automatically stuck. I assume on the right box where f '(x) + g '(x) is where I put the derivatives. So (10x^2)+(8x). But I don't know how to go further. Please help! [FONT=MathJax_Math]f[/FONT]

 

[Deleted member 4993]

f(x)=5x^3 and g(x)= 4x^2, prove that the derivative of the sum of the functions is equal to the sum of the derivative of each function:

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So I don't know how to do this. I start off with getting the derivative of fx and gx so fx is 10x^2 (NO!) and 8x and then I am automatically stuck. I assume on the right box where f '(x) + g '(x) is where I put the derivatives. So (10x^2)+(8x). But I don't know how to go further. Please help! [FONT=MathJax_Math]f[/FONT]

\(\displaystyle \displaystyle{\dfrac{d}{dx}x^n \ = \ n * x^{(n-1)}}\)

 

[pka]

f(x)=5x^3 and g(x)= 4x^2, prove that the derivative of the sum of the functions is equal to the sum of the derivative of each function:

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Let \(\displaystyle S(x)=f(x)+g(x)\) then
\(\displaystyle \begin{align*}S(x+h)-S(x)&=[f(x+h)+g(x+h)]-[f(x)+g(x)]\\&=[f(x+h)-f(x)]+[g(x+h)-g(x)]\end{align*}\)

What is \(\displaystyle \displaystyle{\lim _{h \to 0}}\frac{{S(x + h) - S(x)}}{h}=~?\)

 

[stapel]

Given f(x)=5x^3 and g(x)= 4x^2, prove that the derivative of the sum of the functions is equal to the sum of the derivative of each function:

. . . . .\(\displaystyle \left[\, f(x)\, +\, g(x)\, \right]'\, =\, f'(x)\, +\, g'(x)\)

Are you supposed to "prove from the definition" (that is, by using limits, etc), or just by plug-n-chug? If the latter, then:

a) Find f(x) + g(x).

b) Differentiate the expression in (a). (This is the left-hand side of the derivative equation.)

c) Differentiate f(x).

d) Differentiate g(x).

e) Add the expression in (c) to the expression in (d). (This is the right-hand side of the derivative equation.)

f) Compare the result of (e) with the result of (b).

If the results are the same, then you've shown that the equation is true (at least for these two particular functions).