The Basic Rules

Learning Outcomes

  • State the constant, constant multiple, and power rules
  • Apply the sum and difference rules to combine derivatives

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. We restate this rule in the following theorem.

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]

 

Example: Applying the Constant Rule

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

Try It

Find the derivative of [latex]g(x)=-3[/latex].

The Power Rule

We have 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].

Example: Differentiating [latex]x^3[/latex]

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

Try It

Find [latex]\frac{d}{dx}(x^4)[/latex]

As we shall see, the procedure for finding the derivative of the general form [latex]f(x)=x^n[/latex] is very similar. 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 of [latex]x^2[/latex] in the derivative and the power on [latex]x[/latex] in the derivative decreases by 1. 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 integer powers. Later, we will see that this rule may also be extended first to rational powers of [latex]x[/latex] and then to arbitrary powers of [latex]x[/latex]. Be aware, however, that this rule does not apply to functions in which 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]

Example: Applying the Power Rule

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

Try It

Find the derivative of [latex]f(x)=x^7[/latex].

Watch the following video to see the worked solution to the above Try It.

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.

Sum, Difference, and Constant Multiple Rules

Let [latex]f(x)[/latex] and [latex]g(x)[/latex] be differentiable functions and [latex]k[/latex] be a constant. Then each of the following equations holds.

 

Sum Rule: The derivative of the sum of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the sum of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex].

[latex]\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x))[/latex];

 

that is,

for [latex]j(x)=f(x)+g(x), \, j^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x)[/latex]

 

Difference Rule: The derivative of the difference of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the difference of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex].

[latex]\frac{d}{dx}(f(x)-g(x))=\frac{d}{dx}(f(x))-\frac{d}{dx}(g(x))[/latex];

 

that is,

for [latex]j(x)=f(x)-g(x), \, j^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x)[/latex]

 

Constant Multiple Rule: The derivative of a constant [latex]k[/latex] multiplied by a function [latex]f[/latex] is the same as the constant multiplied by the derivative:

[latex]\frac{d}{dx}(kf(x))=k\frac{d}{dx}(f(x))[/latex];

 

that is,

for [latex]j(x)=kf(x), \, j^{\prime}(x)=kf^{\prime}(x)[/latex]

 

Proof

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

For differentiable functions [latex]f(x)[/latex] and [latex]g(x)[/latex], we set [latex]j(x)=f(x)+g(x)[/latex]. Using the limit definition of the derivative we have

[latex]j^{\prime}(x)=\underset{h\to 0}{\lim}\dfrac{j(x+h)-j(x)}{h}[/latex]

 

By substituting [latex]j(x+h)=f(x+h)+g(x+h)[/latex] and [latex]j(x)=f(x)+g(x)[/latex], we obtain

[latex]j^{\prime}(x)=\underset{h\to 0}{\lim}\dfrac{(f(x+h)+g(x+h))-(f(x)+g(x))}{h}[/latex]

 

Rearranging and regrouping the terms, we have

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

 

We now apply the sum law for limits and the definition of the derivative to obtain

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

[latex]_\blacksquare[/latex]

Example: Applying the Constant Multiple Rule

Find the derivative of [latex]g(x)=3x^2[/latex] and compare it to the derivative of [latex]f(x)=x^2[/latex].

Example: Applying Basic Derivative Rules

Find the derivative of [latex]f(x)=2x^5+7[/latex].

Try It

Find the derivative of [latex]f(x)=2x^3-6x^2+3[/latex].

Watch the following video to see the worked solution to the above Try It.

Example: Finding the Equation of a Tangent Line

Find the equation of the line tangent to the graph of [latex]f(x)=x^2-4x+6[/latex] at [latex]x=1[/latex].

Watch the following video to see the worked solution to Example: Finding the Equation of a Tangent Line.

Try It

Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[/latex] at [latex]x=2[/latex]. Use the point-slope form.

Try It