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

Product and Quotient Rules

Calculus

3. Techniques of Differentiation

Product and Quotient Rules

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3:29 minutes

Problem 3.23

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x⁻¹/² sec (2x)²

Verified step by step guidance

Identify the function components: The given function is 𝔂 = x^-1/2 sec((2x)²). This is a product of two functions: u(x) = x^-1/2 and v(x) = sec((2x)²).

Apply the product rule: The derivative of a product of two functions u(x) and v(x) is given by (uv)' = u'v + uv'.

Differentiate u(x): The derivative of u(x) = x^-1/2 is u'(x) = -1/2 * x^-3/2.

Differentiate v(x): To find v'(x), use the chain rule. The derivative of sec(z) is sec(z)tan(z), where z = (2x)². First, find the derivative of z with respect to x, which is dz/dx = 4x. Then, v'(x) = sec((2x)²)tan((2x)²) * 4x.

Combine the results: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula to find the derivative of 𝔂. Simplify the expression to obtain the final derivative.

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Key Concepts

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

Derivative Rules

Understanding the rules of differentiation, such as the product rule, quotient rule, and chain rule, is essential for finding derivatives of complex functions. The product rule is used when differentiating products of functions, while the chain rule is necessary for composite functions. Mastery of these rules allows for systematic and accurate differentiation.

Trigonometric Functions

The function sec(2x) is a trigonometric function, specifically the secant function, which is the reciprocal of the cosine function. Knowing the derivatives of trigonometric functions, such as sec(x), is crucial for differentiating expressions involving them. The derivative of sec(x) is sec(x)tan(x), which will be applied in this context.

Power Rule

The power rule is a fundamental concept in calculus that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is particularly useful for differentiating functions with exponents, such as x^(-1/2) in the given function. Applying the power rule correctly is vital for simplifying the differentiation process.