The derivative of a linear combination of functions is the linear combination of the individual derivatives. In other words, the derivative of the sum is the sum of the derivatives, and scaling a function by a constant multiplies the derivative by the same constant.
If f is multiplied by the constant k, then the numerator of the difference quotient, kf(x+h)-kf(x), is k times as large, hence the limit, as h approaches 0, is multiplied by k.
Similarly, the numerator produced by the function f+g can be rearranged to give f(x+h)-f(x) + g(x+h)-g(x). Since the limit of the sum is the sum of the limits, we can simply add the derivatives together.
Combine these results, and the derivative of a linear combination of functions is the linear combination of the derivatives.