Differentiation Rules: Learn It 1

  • Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers
  • Apply specific rules to find derivatives of functions multiplied or divided by each other
  • Use a combination of rules to calculate derivatives for polynomial and rational functions

The Basic Rules

Finding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process. In this section, we develop rules for finding derivatives that allow us to bypass this process. We begin with the basics.

The functions [latex]f(x)=c[/latex] and [latex]g(x)=x^n[/latex] where [latex]n[/latex] is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.

The Constant Rule

We first apply the limit definition of the derivative to find the derivative of the constant function, [latex]f(x)=c[/latex]. For this function, both [latex]f(x)=c[/latex] and [latex]f(x+h)=c[/latex], so we obtain the following result:

[latex]\begin{array}{ll}f^{\prime}(x) & =\underset{h\to 0}{\lim}\dfrac{f(x+h)-f(x)}{h} \\ & =\underset{h\to 0}{\lim}\dfrac{c-c}{h} \\ & =\underset{h\to 0}{\lim}\dfrac{0}{h} \\ & =\underset{h\to 0}{\lim}0=0 \end{array}[/latex]

The rule for differentiating constant functions is called the constant rule. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is 0. 

the constant rule

Let [latex]c[/latex] be a constant.

If [latex]f(x)=c[/latex], then [latex]f^{\prime}(c)=0[/latex]

Alternatively, we may express this rule as

[latex]\dfrac{d}{dx}(c)=0[/latex]

Find the derivative of [latex]f(x)=8[/latex].

The Power Rule

We have previously shown that,

[latex]\dfrac{d}{dx}\left(x^2\right)=2x[/latex]   and   [latex]\dfrac{d}{dx}\left(x^{\frac{1}{2}}\right)=\dfrac{1}{2}x^{−\frac{1}{2}}[/latex]

At this point, you might see a pattern beginning to develop for derivatives of the form [latex]\frac{d}{dx}(x^n)[/latex].

We continue our examination of derivative formulas by differentiating power functions of the form [latex]f(x)=x^n[/latex] where [latex]n[/latex] is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers.

Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, [latex]\frac{d}{dx}(x^3)[/latex].

Find [latex]\frac{d}{dx}(x^3)[/latex]

As we shall see, the procedure for finding the derivative of the general form [latex]f(x)=x^n[/latex] is straightforward. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate [latex]f(x)=x^3[/latex], the power on [latex]x[/latex] becomes the coefficient, and the new exponent decreases by [latex]1[/latex], resulting in [latex]\frac{d}{dx}(x^3)=3x^2[/latex]

The following theorem states that this power rule holds for all positive integer powers of [latex]x[/latex]. We will eventually extend this result to negative and rational powers of [latex]x[/latex]. 

Be aware that this rule does not apply to functions where a constant is raised to a variable power, such as [latex]f(x)=3^x[/latex].

the power rule

Let [latex]n[/latex] be a positive integer. If [latex]f(x)=x^n[/latex], then

[latex]f^{\prime}(x)=nx^{n-1}[/latex]

 

Alternatively, we may express this rule as

[latex]\dfrac{d}{dx}(x^n)=nx^{n-1}[/latex]

Proof

For [latex]f(x)=x^n[/latex] where [latex]n[/latex] is a positive integer, we have

[latex]f^{\prime}(x)=\underset{h\to 0}{\lim}\frac{(x+h)^n-x^n}{h}[/latex].

 

Since [latex](x+h)^n=x^n+nx^{n-1}h+\binom{n}{2}x^{n-2}h^2+\binom{n}{3}x^{n-3}h^3+\cdots+nxh^{n-1}+h^n[/latex],

we see that

[latex](x+h)^n-x^n=nx^{n-1}h+\binom{n}{2}x^{n-2}h^2+\binom{n}{3}x^{n-3}h^3+\cdots+nxh^{n-1}+h^n[/latex]

 

Next, divide both sides by [latex]h[/latex]:

[latex]\large \frac{(x+h)^n-x^n}{h}=\frac{nx^{n-1}h+\binom{n}{2}x^{n-2}h^2+\binom{n}{3}x^{n-3}h^3+\cdots+nxh^{n-1}+h^n}{h}[/latex]

 

Thus,

[latex]\frac{(x+h)^n-x^n}{h}=nx^{n-1}+\binom{n}{2}x^{n-2}h+\binom{n}{3}x^{n-3}h^2+\cdots+nxh^{n-2}+h^{n-1}[/latex]

 

Finally,

[latex]\begin{array}{ll}f^{\prime}(x) & =\underset{h\to 0}{\lim}(nx^{n-1}+\binom{n}{2}x^{n-2}h+\binom{n}{3}x^{n-3}h^2+\cdots+nxh^{n-1}+h^n) \\ & =nx^{n-1} \end{array}[/latex]

[latex]_\blacksquare[/latex]

Find the derivative of the function [latex]f(x)=x^{10}[/latex] by applying the power rule.