Developing Derivative Properties

Section 2.4 Developing Derivative Properties

Standards

Calculate polynomial derivatives
Calculate trigonometric derivatives
Calculate exponential derivatives
Calculate derivatives with the product rule
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 2.4.1. Calculate the derivative of \(f(x)=x\).

The derivative of \(f(x)=x\) at any point is \(f^\prime(x)=1.\)

Solution.

\begin{align*} \lim_{x \to a} \frac{f(x)-f(a)}{x-a} = \amp \text{ definition}\\ \lim_{x \to a} \frac{x-a}{x-a} = \amp \text{ substituting the function}\\ \lim_{x \to a} 1 = \amp \text{ algebra}\\ 1. \amp \text{ known limit} \end{align*}

Example 2.4.2. Calculate the derivative of \(f(x)=k\).

The derivative of \(f(x)=k\) at any point is \(f^\prime(x)=0.\)

Solution.

\begin{align*} \lim_{x \to a} \frac{f(x)-f(a)}{x-a} = \amp \text{ definition}\\ \lim_{x \to a} \frac{k-k}{x-a} = \amp \text{ substituting the function}\\ \lim_{x \to a} 0 = \amp \text{ algebra}\\ 0. \amp \text{ known limit} \end{align*}

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.

Instructions.

Use the slide labeled da to view the limit as it approaches zero giving us the derivative.

Figure 2.4.4. Product Rule as Change in Area

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?

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