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### Differential Sum Rule Demonstrated

##### Contents

The sum rule of differentiation is that the derivative of the sums of two functions is the sum of the derivatives of each of the functions.

{$${d \over dx} (f(x) + g(x)) = {d \over dx} f(x) + {d \over dx} g(x)$$}

The definition of the derivative is:

{$$k'(x) = \lim_{ h \rightarrow0} {{k(x + h) -k(x)} \over { h}}$$}

The function name has been changed to k to avoid a collision.

Now,

{\begin{align} k(x) &= f(x) + g(x), \quad \text{ So:} \cr k'(x) &= \lim_{ h \rightarrow0} {{f(x + h) + g(x+h)-(f(x)+g(x))} \over { h}} \cr k(x) &= \lim_{ h \rightarrow0} {{f(x + h) -f(x) + g(x+h)-g(x)} \over { h}} \cr k;(x) &= \lim_{ h \rightarrow0} {{f(x + h) -f(x)} \over h} + \lim_{ h \rightarrow0} {{g(x+h)-g(x)} \over { h}} \cr \end{align}}

But, by definition,

{\begin{align} \lim_{ h \rightarrow0} {{f(x + h) -f(x)} \over h} &= f'(x), \text{ and} \cr \lim_{ h \rightarrow0} {{g(x+h)-g(x)} \over { h}} &= g'(x), \text{ so} \cr k'(x) &= f'(x) + g'(x) \end{align}}

which is what was required, albeit it in a different notation.

Sources:

Recommended:

Category: Math Calculua

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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### December 23, 2018

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