# Derivative Patterns

Site: | Saylor Academy |

Course: | MA005: Calculus I |

Book: | Derivative Patterns |

Printed by: | Guest user |

Date: | Tuesday, August 6, 2024, 1:47 PM |

## Description

Read this section to learn about patterns of derivatives. Work through practice problems 1-8.

Polynomials are very useful, but they are not the only functions we need. This section uses the ideas of the two previous sections to develop techniques for differentiating powers of functions, and to determine the derivatives of some particular functions which occur often in applications, the trigonometric and exponential functions.

As you focus on learning how to differentiate different types and combinations of functions, it is important to remember what derivatives are and what they measure. Calculators and personal computers are available to calculate derivatives. Part of your job as a professional will be to decide which functions need to be differentiated and how to use the resulting derivatives. You can succeed at that only if you understand what a derivative is and what it measures.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.4-More-Differentiation-Problems.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.

If we apply the Product Rule to the product of a function with itself, a familiar pattern emerges.

**Practice 1:** What is the pattern here? What do you think the results will be for and ?

We could keep differentiating higher and higher powers of by writing them as products of lower powers of and using the Product Rule, but the Power Rule For Functions guarantees that the pattern we just saw for the small integer powers also works for all constant powers of functions.

**Power Rule For Functions:** If is any constant,

The Power Rule for Functions is a special case of a more general theorem, the Chain Rule, which we will examine in Section 2.4. The Power Rule For Functions will be proved after the Chain Rule.

**Example 1:** Use the Power Rule for Functions to find

Solution: (a) To match the pattern of the Power Rule for , let and .

(b) To match the pattern for , we can let and take . Then

(c) To match the pattern for , Let and . Then

**Practice 2:** Use the Power Rule for Functions to find

**Example 2:** Use calculus to show that the line tangent to the circle at the point has slope .

Solution: The top half of the circle is the graph of so

As a check, you can verify that the slope of the radial line through the center of the circle and the point has slope and is perpendicular to the tangent line which has a slope of .

We have some general rules which apply to any elementary combination of differentiable functions, but in order to use the rules we still need to know the derivatives of each of the particular functions. Here we will add to the list of functions whose
derivatives we know.

##### Derivatives of the Trigonometric Functions

We know the derivatives of the sine and cosine functions, and each of the other four trigonometric functions is just a ratio involving sines or cosines. Using the Quotient Rule, we can differentiate the rest of the trigonometric functions.

Proof: From trigonometry we know , and and we know and Using the Quotient Rule,

Instead of the Quotient Rule, we could have used the Power Rule to calculate .

**Practice 3:** Use the Quotient Rule on to prove that f .

**Practice 4:** Prove that . The justification of this result is very similar to the justification for

**Practice 5:** Find (a) , (b) and (c) .

##### Derivative of

We can use graphs of exponential functions to estimate the slopes of their tangent lines or we can numerically approximate the slopes.

**Example 3:** Estimate the derivative of at the point by approximating the slope of the line tangent to at that point

Solution: We can get estimates from the graph of by carefully graphing for small values of , sketching secant lines, and then measuring the slopes of the secant lines (Fig. 1).

We can also find the slope numerically by using the definition of the derivative,

, and evaluating for some very small values of .

0.1 | 0.717734625 | ||

-0.I | 0.669670084 | ||

0.01 | 0.69555 | ||

-0.01 | 0.690750451 | ||

0.001 | 0.6933874 | ||

-0.001 | 0.69290695 | ||

From the table we can see that .

**Practice 6:** Fill in the table for , and show that the slope of the line tangent to at is approximately . (Fig. 2)

At , the slope of the tangent to is less than 1, and the slope of the tangent to is slightly greater than 1. (Fig. 3) There is a number, denoted , between 2 and 3 so that the slope of the tangent to is exactly 1: The number .

is irrational and is very important and common in calculus and applications.

Once we grant that , it is relatively straightforward to calculate .

Proof:

The function is its own derivative: . The height of at any point and the slope of the tangent to at that point are the same: as the graph gets higher, its slope gets steeper.

**Example 4:** Find (a) , (b) and (c)

Solution: (a) Using the Product Rule with and ,

(b) Using the Quotient Rule with and ,

(c) Using the Power Rule for Functions with and ,

The derivative of a function is a new function , and we can calculate the derivative of this new function to get the derivative of the derivative of , denoted by and called the second derivative of . For example, if then and .

**Definitions:** The first derivative of is , the rate of change of .

The second derivative of is , the rate of change of . The third derivative of is , the rate of change of ".

**Practice 8:** Find , and for , and

If represents the position of a particle at time , then will represent the velocity (rate of change of the position) of the particle and will represent the acceleration (the rate of change of the velocity) of the particle.

**Example 5:** The height (feet) of a particle at time seconds is . Find the height, velocity and acceleration of the particle when , and seconds.

Solution: so feet, feet, and feet

The velocity is so , and . At each of these times the velocity is positive and the particle is moving upward, increasing in height.

We will examine the geometric meaning of the second derivative later.

In Section 1.2 we saw that the "holey" function

is discontinuous at every value of , so at every is not differentiable. We can create graphs of continuous functions that are not differentiable at several places just by putting corners at those places, but how many corners can a continuous function have? How badly can a continuous function fail to be differentiable?

In the midâ€“1800s, the German mathematician Karl Weierstrass surprised and even shocked the mathematical world by creating a function which was **continuous everywhere but differentiable nowhere** - a function whose graph was everywhere connected and everywhere
bent! He used techniques we have not investigated yet, but we can start to see how such a function could be built.

Start with a function (Fig. 4) which zigzags between the values and and has a "corner" at each integer. This starting function is continuous everywhere and is differentiable everywhere except at the integers. Next create a list of functions , each of which is a lot shorter but with many more "corners" than the previous ones. For example, we might make zigzag between the values and and have "corners" at , , etc., and zigzag between and and have "corners" at , , etc. If we add and , we get a continuous function (since the sum of two continuous functions is continuous) which will have corners at , If we then add to the previous sum, we get a new continuous function with even more corners. If we continue adding the functions in our list "indefinitely", the final result will be a continuous function which is differentiable nowhere.

We haven't developed enough mathematics here to precisely describe what it means to add an infinite number of functions together or to verify that the resulting function is nowhere differentiable, but we will. You can at least start to imagine what a strange, totally "bent" function it must be.

Until Weierstrass created his "everywhere continuous, nowhere differentiable" function, most mathematicians thought a continuous function could only be "bad" in a few places, and Weierstrass' function was (and is) considered "pathological", a great example of how bad something can be. The mathematician Hermite expressed a reaction shared by many when they first encounter Weierstrass' function:

"I turn away with fright and horror from this lamentable evil of functions which do not have derivatives".

##### IMPORTANT RESULTS

**Derivatives of the Trigonometric Functions:**

**Practice 1: **The pattern is and

**Practice 3:**

**Practice 4:**

**Practice 6:**

**Practice 8:**