Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

Derivatives of Trig Functions

Calculus

3. Techniques of Differentiation

Derivatives of Trig Functions

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2:26 minutes

Problem 3.5.29

Textbook Question

Derivatives

In Exercises 27–32, find dp/dq.

p = (sin q + cos q) / cos q

Verified step by step guidance

Step 1: Start by identifying the function p in terms of q. Here, p is given as \( p = \frac{\sin q + \cos q}{\cos q} \).

Step 2: Simplify the expression for p. Divide each term in the numerator by the denominator: \( p = \frac{\sin q}{\cos q} + \frac{\cos q}{\cos q} \). This simplifies to \( p = \tan q + 1 \).

Step 3: Differentiate p with respect to q. The derivative of \( \tan q \) with respect to q is \( \sec^2 q \), and the derivative of a constant (1) is 0.

Step 4: Combine the derivatives to find \( \frac{dp}{dq} \). Since \( p = \tan q + 1 \), \( \frac{dp}{dq} = \sec^2 q + 0 \).

Step 5: Conclude that the derivative of p with respect to q is \( \frac{dp}{dq} = \sec^2 q \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate at which a function changes as its input changes. It is a fundamental concept in calculus that measures how a function's output value varies with respect to changes in its input variable. The derivative of a function can be interpreted as the slope of the tangent line to the function's graph at a given point.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule allows us to differentiate it by multiplying the derivative of the outer function by the derivative of the inner function. This is essential when dealing with functions that are expressed in terms of other functions, as in the case of p = (sin q + cos q) / cos q.

Trigonometric Derivatives

Trigonometric derivatives involve the differentiation of functions that include trigonometric functions such as sine and cosine. The derivatives of these functions are well-defined: the derivative of sin(q) is cos(q), and the derivative of cos(q) is -sin(q). Understanding these derivatives is crucial for solving problems involving trigonometric functions, especially when applying the quotient rule in this context.

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In Exercises 1–18, find dy/dx.

y = (sec x + tan x)(sec x − tan x)

Textbook Question

Derivatives

In Exercises 1–18, find dy/dx.

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Textbook Question

Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ

Textbook Question

Derivatives

In Exercises 23–26, find dr/dθ.

r = (1 + sec θ) sin θ

Textbook Question

Derivatives

In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)

Textbook Question

Consider the function

f(x) = { x² cos(2/x), x ≠ 0

0, x = 0

b. Determine f' for x ≠ 0.

Textbook Question

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

b. Find the spring’s velocity when t = 0, t = π/3, and t = 3π/4.

Textbook Question

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

b. Find the particle’s velocity when t = 0, t = π/2, and t = π.

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