Subject: Calculus

Sum Rule In Differentiation


When a function is expressed as a sum of two other functions, then the derivative of that function is also the sum of the individual derivatives of the other functions. This process is what we call the sum rule in differentiation.

Rule #2: (Derivative of a Sum)


If h and g are differentiable at x and f(x)=h(x)+g(x), then the derivative f(x) is f'(x)=g'(x)+h'(x).

While the above theorem presents the sum of only two functions g and h, this theorem is also applicable in finding the derivative for infinite sum of functions.

Derivative of the subtraction of two or more functions.

Maybe you would ask me why is there no rule on the derivative of the difference of two or more functions. Well, rule #2 above also covers to this kinds of derivative. Remember that subtraction also involves the process of addition. So when we have a difference of two or more functions, its derivative is also the difference of the individual derivatives of the two functions or more.

Example #1

Find the derivative of f(x), where f(x)=2x+x^{2}.

Solution: We know that the derivative of 2x and x^{2} are 2 and 2x. Thus, the derivative of the sum of 2x and x^{2}, according to Rule #2 is just the sum of the individual derivatives.
Thus, f'(x)=2+2x=2(x+1).

Example #2

Find the derivative of f(x), where f(x)=2x-x^{2}-\sin{x}+\cos{x}.

Solution: We know that the derivative of 2x and x^{2} are 2 and 2x and the derivative of \sin{x} and \cos{x} is -\sin{x} and \cos{x}. Thus, f'(x)=2(x+1)-\cos{x}-\sin{x}.

Example #3

Find the derivative of f(x), where f(x)=1+2+3+ x^2-x.

Solution: We know that the derivative of 1, 2 and 3 are 0; the derivative of x^2 is2x and the derivative of x is 1, thus f'(x)=2x-1.

Example #4

Find the derivative of f(x), where f(x)=e^{2x}-\cos{x}.

Solution: We know that the derivative of e^{2x} is 2e^{2x} and the derivative of \cos{x} is -\sin{x}. Thus, f'(x)=2e^{2x}+\sin{x}.

Example #5

Find the derivative of f(x), where f(x)= 3+g(x).

Solution: We know that the derivative of 3 is 0 and the derivative of g(x) is g'(x). Thus, f'(x)=g'(x).

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