Differentiating Polynomials: Power, Constant Multiple, Sum, and Difference Rules

From Limits to Rules: A More Efficient Path

Calculating every derivative from the limit definition, $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, is impractical. As stated in your textbook, "it would be tedious if we always had to use the definition". To that end, mathematicians have developed a set of rules that streamline this process for common types of functions. We'll begin with the simplest cases and build from there.

The following comprehensive video from the "Understand the Math" series walks through all the rules for this lesson. We will refer back to it as we introduce each rule. Feel free to watch it in its entirety or use it as a reference as we proceed.

Rule 1: The Derivative of a Constant Function

Let's start with the simplest function: $f(x)=c$, where $c$ is a constant. The graph of this function is a horizontal line. Since the slope of a horizontal line is always zero, it follows that its derivative must be zero.

The Constant Rule: If $c$ is a constant, then

This is simple to prove using the limit definition, as shown in the first minute of the video.

Rule 2: The Power Rule

The Power Rule is arguably the most important rule for differentiating polynomials. It provides a simple formula for the derivative of $f(x)=x^n$.

The Power Rule (General Version): If $n$ is any real number, then

In words: bring the exponent down as a coefficient and then subtract one from the exponent.

For example:

• If $f(x)=x^4$, then $f^{\prime }(x)=4x^{4-1}=4x^3$.
• If $y=x^{100}$, then $y^{\prime }=100x^{99}$.

Initially, this rule was proven for positive integers, but it works for all real numbers, including negative and fractional exponents. This is incredibly powerful because it allows us to differentiate functions like $f(x)=\frac{1}{x^2}$ and $g(x)=\sqrt[3]{x}$ by first rewriting them as powers of $x$.

• $f(x)=\frac{1}{x^2}=x^{-2}$, so $f^{\prime }(x)=-2x^{-2-1}=-2x^{-3}=-\frac{2}{x^3}$.
• $g(x)=\sqrt[3]{x}=x^{1/3}$, so $g^{\prime }(x)=\frac{1}{3}{x}^{(1/3)-1}=\frac{1}{3}{x}^{-2/3}$.

The video also provides a great walkthrough of these concepts.

Rules 3 & 4: Linearity of the Derivative

The next two rules, the Constant Multiple Rule and the Sum Rule, allow us to handle more complex functions. Together, they establish that differentiation is a linear operation. This is a concept you may have encountered in other areas of mathematics and computer science.

The Constant Multiple Rule: If $c$ is a constant and $f$ is a differentiable function, then

This rule says you can "pull out" a constant coefficient before differentiating. For example, to find the derivative of $5x^4$, we can do:

The Sum and Difference Rules: If $f$ and $g$ are both differentiable, then

This rule states that the derivative of a sum is the sum of the derivatives. This is what allows us to differentiate a polynomial term by term.

Putting It All Together: Differentiating Polynomials

With these four rules in hand, we can now differentiate any polynomial function. The strategy is to use the Sum/Difference rules to break the polynomial into individual terms, use the Constant Multiple rule to handle coefficients, and use the Power Rule to differentiate each power of $x$.

Consider $P(x)=8x^7+12x^5-4x^4-6x+5$. Let's find $P^{\prime }(x)$.

Often, you'll need to perform some algebraic simplification before you can apply the rules. The following video provides excellent examples of this.

Application: Finding Horizontal Tangents

Let's tie this new algebraic power back to the geometry of the derivative. In the last lesson, you learned that a horizontal tangent on the graph of $f(x)$ corresponds to a zero on the graph of $f^{\prime }(x)$. Now, we can find the exact locations of these points algebraically. A horizontal tangent occurs where the slope is zero, meaning we need to find where the derivative is zero.

Example: Find the points on the curve $y=x^4-6x^2+4$ where the tangent line is horizontal.

1. Find the derivative:
   Using our new rules, we differentiate term by term:

   $$\frac{dy}{dx}=\frac{d}{dx}(x^4)-6\frac{d}{dx}(x^2)+\frac{d}{dx}(4)=4x^3-12x+0=4x^3-12x$$

2. Set the derivative to zero and solve:
   We need to find $x$ such that $\frac{dy}{dx}=0$.

   $$4x^3-12x=0$$ $$4x(x^2-3)=0$$

   This gives us three solutions: $x=0$, $x=\sqrt{3}$, and $x=-\sqrt{3}$.

3. Find the corresponding y-coordinates:
   We plug these x-values back into the original equation $y=x^4-6x^2+4$.

   • If $x=0$, $y=(0{)}^4-6(0{)}^2+4=4$. Point: $(0,4)$.
   • If $x=\sqrt{3}$, $y=(\sqrt{3}{)}^4-6(\sqrt{3}{)}^2+4=9-6(3)+4=-5$. Point: $(\sqrt{3},-5)$.
   • If $x=-\sqrt{3}$, $y=(-\sqrt{3}{)}^4-6(-\sqrt{3}{)}^2+4=9-18+4=-5$. Point: $(-\sqrt{3},-5)$.

So, the curve has horizontal tangents at the points $(0,4)$, $(\sqrt{3},-5)$, and $(-\sqrt{3},-5)$.

The Asher Roberts video you looked at earlier concludes with this exact example. You can watch that segment for a final reinforcement of this process.

Conclusion

You have now acquired a fundamental set of tools for differentiation. By moving beyond the limit definition, you can now compute derivatives of polynomial functions quickly and mechanically.

Key Takeaways:

• Constant Rule: The derivative of a constant is 0.
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$. This is your main tool.
• Constant Multiple Rule: $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$. Constants can be factored out.
• Sum/Difference Rule: $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$. You can differentiate term by term.
• These rules combine to allow the efficient differentiation of any polynomial.
• Sometimes, algebraic manipulation (expanding, distributing, simplifying) is required before the rules can be applied.

In our next lesson, we will expand our toolkit. What happens when our function is a product of two functions, like $f(x)=(x^2+1)(x^3-4)$, or a quotient? The Sum Rule doesn't apply here, and a common mistake is to assume the derivative of a product is the product of the derivatives (it is not!). We will need two new, powerful rules: the Product Rule and the Quotient Rule.