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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2:56 minutes

Problem 3.5.17

Textbook Question

Derivatives

In Exercises 1–18, find dy/dx.

f(x) = x³ sin x cos x

Verified step by step guidance

Identify the function f(x) = x³ sin(x) cos(x) and recognize that it is a product of three functions: u(x) = x³, v(x) = sin(x), and w(x) = cos(x).

Apply the product rule for derivatives, which states that if you have a product of functions u(x), v(x), and w(x), then the derivative is given by: (uvw)' = u'vw + uv'w + uvw'.

Calculate the derivative of each individual function: u'(x) = 3x², v'(x) = cos(x), and w'(x) = -sin(x).

Substitute these derivatives into the product rule formula: dy/dx = (3x²)(sin(x))(cos(x)) + (x³)(cos(x))(cos(x)) + (x³)(sin(x))(-sin(x)).

Simplify the expression by combining like terms and using trigonometric identities if necessary to express the derivative in its simplest form.

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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 of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The notation dy/dx indicates the derivative of y with respect to x, and it can be computed using various rules such as the power rule, product rule, and chain rule.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by d(uv)/dx = u'v + uv', where u' and v' are the derivatives of u and v, respectively. This rule is essential for differentiating functions that are products of simpler functions, such as the given function f(x) = x³ sin x cos x.

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental in calculus, especially when dealing with derivatives. These functions have specific derivatives: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is crucial when applying the product rule to functions that involve trigonometric components, as seen in the function f(x) = x³ sin x cos x.

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Related Practice

Textbook Question

{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?

f(x) = sin x on [−π,π]

Textbook Question

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>

b. Which runner has the greater average angular velocity?

Textbook Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/4 cot x−1 / x−π/4

Textbook Question

Derivatives

In Exercises 1–18, find dy/dx.

y = (sec x + tan x)(sec x − tan x)

Textbook Question

Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ