3.3 Differentiation rules  (Page 2/14)

As we shall see, the procedure for finding the derivative of the general form $f\left(x\right)=x^n$ is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate $f\left(x\right)=x^3,$ the power on $x$ becomes the coefficient of $x^2$ in the derivative and the power on $x$ in the derivative decreases by 1. The following theorem states that the power rule    holds for all positive integer powers of $x.$ We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of $x$ and then to arbitrary powers of $x.$ Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as $f(x)=3^x.$

Proof

For $f\left(x\right)=x^n$ where $n$ is a positive integer, we have

we see that

Next, divide both sides by h :

Thus,

Finally,

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The sum, difference, and constant multiple rules

We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant. These rules are summarized in the following theorem.

Proof

We provide only the proof of the sum rule here. The rest follow in a similar manner.

For differentiable functions $f\left(x\right)$ and $g\left(x\right),$ we set $j\left(x\right)=f\left(x\right)+g\left(x\right).$ Using the limit definition of the derivative we have

By substituting $j\left(x+h\right)=f\left(x+h\right)+g\left(x+h\right)$ and $j\left(x\right)=f\left(x\right)+g\left(x\right),$ we obtain

Rearranging and regrouping the terms, we have