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

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

Previous problemNext problem

2:18 minutes

Problem 3.15

Textbook Question

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

s = (sec t + tan t)⁵

Verified step by step guidance

Identify the function: We have s = (sec t + tan t)⁵. This is a composite function where the outer function is u⁵ and the inner function is u = sec t + tan t.

Apply the chain rule: The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x). Here, f(u) = u⁵ and g(t) = sec t + tan t.

Differentiate the outer function: The derivative of u⁵ with respect to u is 5u⁴. So, f'(u) = 5(sec t + tan t)⁴.

Differentiate the inner function: The derivative of sec t is sec t tan t, and the derivative of tan t is sec² t. Therefore, g'(t) = sec t tan t + sec² t.

Combine the derivatives using the chain rule: Multiply the derivative of the outer function by the derivative of the inner function: s'(t) = 5(sec t + tan t)⁴ * (sec t tan t + sec² t).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule, depending on the form of the function.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as seen in the given problem.

Trigonometric Functions

Trigonometric functions, such as secant (sec) and tangent (tan), are fundamental functions in calculus that relate angles to ratios of sides in right triangles. Their derivatives are essential for solving problems involving rates of change in contexts like physics and engineering. Understanding the derivatives of these functions is crucial for applying the chain rule effectively in the given exercise.