Section 2.4 Developing Derivative Properties
Standards
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Calculate polynomial derivatives
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Calculate trigonometric derivatives
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Calculate exponential derivatives
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Calculate derivatives with the product rule
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Calculate derivatives with the quotient rule
Just like with limits we can calculate derivatives by breaking them down using arithmetic and algebraic properties.
Our emphasis will be on calculating derivatives, though two of the rules make important statements about the relationship between rates in applications. Calculating derivatives is a skill useful in recognizing relationships in other classes, though because of computer it is no longer used in jobs.
Subsection 2.4.1 Calculating Derivatives Using the Definition
The definition derives from the concepts. Using it to calculate derivatives takes some creativity.
Example 2.4.2. Calculate the derivative of \(f(x)=k\).
Example 2.4.3. Calculate the derivative of \(f(x)=3x^2-1\).
Calculate the derivative of \(f(x)=3x^2-1\text{.}\)
Solution.
\begin{align*}
\lim_{x \to a} \frac{f(x)-f(a)}{x-a} = \amp \text{ definition}\\
\lim_{x \to a} \frac{(3x^2-1)-(3a^2-1)}{x-a} = \amp \text{ substituting the function}\\
\lim_{x \to a} \frac{3x^2-3a^2-1+1}{x-a} = \amp \text{ commutativity}\\
\lim_{x \to a} \frac{3(x^2-a^2)}{x-a} = \amp \text{ factor}\\
\lim_{x \to a} \frac{3(x-a)(x+a)}{x-a} = \amp \text{ factor more}\\
\lim_{x \to a} 3(x+a) = \amp \text{ divide}\\
6a \amp \text{ known limits}
\end{align*}
Subsection 2.4.2 Arithmetic Derivative Properties
Subsubsection 2.4.2.1 Derivative of a Sum
What can we do if we want to calculate the derivative of a sum of functions, i.e., \(f(x)+g(x)\text{?}\) We can try using the definition.
\begin{align*}
\frac{d(f(x)+g(x))}{dx} = \amp \\
\lim_{x \to a} \frac{(f(x)+g(x))-(f(a)-g(a))}{x-a} = \amp \text{ definition}\\
\lim_{x \to a} \frac{(f(x)-f(a))+(g(x)-g(a))}{x-a} = \amp \text{ commutative property}\\
\lim_{x \to a} \frac{f(x)-f(a)}{x-a}+\frac{g(x)-g(a)}{x-a} = \amp \text{ algebraic property}\\
\lim_{x \to a} \frac{f(x)-f(a)}{x-a}+ \lim_{x \to a} \frac{g(x)-g(a)}{x-a} = \amp \text{ limit summation property}\\
\frac{d f(x)}{dx}+ \frac{d g(x)}{dx} = \amp \text{ definition}
\end{align*}
Subsubsection 2.4.2.2 Derivative of a Scalar Product
What can we do if we want to calculate the derivative of a scalar times a function, i.e., \(k \cdot f(x)\text{?}\) Use the definition again.
\begin{align*}
\frac{d k f(x)}{dx} = \amp \\
\lim_{x \to a} \frac{k f(x)- k f(a)}{x-a} = \amp \text{the definition}
\end{align*}
Finish this proof. Remember the example above.
Subsubsection 2.4.2.3 Derivative of a Product
There is also a property for a derivative of a product, but it is not simply applying the limit property for a product. Provide a reason for each line of this proof.
\begin{align*}
\frac{d f(x)g(x)}{dx} = \amp \text{the question}\\
\lim_{x \to a} \frac{f(x)g(x)-f(a)g(a)}{x-a} = \amp \fillinmath{XXX}\\
\lim_{x \to a} \frac{f(x)g(x)-f(x)g(a)+f(x)g(a)-f(a)g(a)}{x-a} = \amp \text{ a very clever step} \\
\lim_{x \to a} \frac{f(x)[g(x)-g(a)]+[f(x)-f(a)]g(a)}{x-a} = \amp \fillinmath{XXX} \\
\lim_{x \to a} \frac{f(x)[g(x)-g(a)]}{x-a} + \frac{[f(x)-f(a)]g(a)}{x-a} = \amp \fillinmath{XXX}\\
\lim_{x \to a} \frac{f(x)[g(x)-g(a)]}{x-a} + \lim_{x \to a} \frac{[f(x)-f(a)]g(a)}{x-a} = \amp \fillinmath{XXX} \\
\lim_{x \to a} f(x)\lim_{x \to a} \frac{g(x)-g(a)}{x-a} + g(a)\lim_{x \to a} \frac{f(x)-f(a)}{x-a} = \amp \fillinmath{XXX}\\
f(a)g'(a)+g(a)f'(a). \amp \fillinmath{XXX}
\end{align*}
We can illustrate the produce rule using a product to represent an area of a rectangle as in Figure 2.4.4 Note that the change in the area of the rectangle is the gnomon (inverted L shape) consisting of \(f(a)g^\prime(a)\text{,}\) \(f^\prime(a)g^\prime(a)\text{,}\) and \(g(a)f^\prime(a)\text{.}\) This matches the product rule \(f(a)g^\prime(a)+g(a)f^\prime(a)\) except for the small square \(f^\prime(a)g^\prime(a)\text{.}\) In the limit this square’s area goes to zero. Indeed in early development of the derivative this is exactly how they did it.
Subsubsection 2.4.2.4 Derivative of a Quotient
The quotient property of calculating derivatives can be proved similarly to the product property. The property is
\begin{equation*}
\frac{d \frac{f(x)}{g(x)}}{dx} = \frac{g(x)f^\prime(x)-f(x)g^\prime(x)}{g^2(x)} \text{.}
\end{equation*}
Subsection 2.4.3 Using Properties to Calculate Derivatives
Subsubsection 2.4.3.1 Derivative of Integer Powers
Checkpoint 2.4.5.
Using the product rule calculate the derivatives of the following functions. \(f_2(x)=x^2\text{,}\) \(f_3(x)=x^3\text{,}\) \(f_4(x)=x^4\text{,}\) \(f_5(x)=x^5\text{.}\) Conjecture the pattern for all integer powers. Explain why the product rule cannot be used to prove this pattern for \(f_\pi(x)=x^\pi.\)
Subsubsection 2.4.3.2 Derivatives of trigonometric functions
We can prove that for \(f(x)=\sin(x)\text{,}\) \(f^\prime(x)=\cos(x).\) Similarly we can show that for \(g(x)=\cos(x)\text{,}\) \(g^\prime(x)=-\sin(x).\)
Checkpoint 2.4.6.
Using trig identities and the derivative properties, find the derivatives for \(\tan(x)\text{,}\) \(\cot(x)\text{,}\) \(\sec(x)\text{,}\) and \(\csc(x)\text{.}\)
Subsubsection 2.4.3.3 Calculating Using Multiple Properties
Many functions involve more than one of the properties. This leads to the question about how to apply more than one property when calculating a derivative.
Example 2.4.7.
We calculate the derivative of \(f(x)=\sin(x)+x^2\cos(x)\text{.}\)
\begin{align*}
f^\prime(x) & = \frac{d \sin(x)}{dx} + \frac{d x^2\cos(x)}{dx} & & \text{sum property}\\
& = \cos(x) + \frac{d x^2\cos(x)}{dx} & & \text{known form}\\
& = \cos(x) + \frac{d x^2}{dx}\cos(x)+x^2\frac{d \cos(x)}{dx} & & \text{product property}\\
& = \cos(x) + 2x\cos(x)+x^2(-\sin(x)) & & \text{known forms}\\
& = \cos(x) + 2x\cos(x)-x^2\sin(x)
\end{align*}
Checkpoint 2.4.8.
Why was the sum property used before the product property in Example 2.4.7? What order is this?
