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

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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1:13 minutes

Problem 7.3.22

Textbook Question

22–36. Derivatives Find the derivatives of the following functions.

f(x) = sinh 4x

Verified step by step guidance

Step 1: Recall the derivative rule for hyperbolic sine functions. The derivative of sinh(u) with respect to x is cosh(u) * du/dx.

Step 2: Identify the inner function u in the given function f(x) = sinh(4x). Here, u = 4x.

Step 3: Compute the derivative of the inner function u = 4x with respect to x. The derivative of 4x is 4.

Step 4: Apply the chain rule. Substitute u = 4x into the derivative formula: f'(x) = cosh(4x) * 4.

Step 5: Simplify the expression to write the derivative as f'(x) = 4 * cosh(4x).

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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 function's graph at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule.

Hyperbolic Functions

Hyperbolic functions, such as sinh (hyperbolic sine) and cosh (hyperbolic cosine), are analogs of the trigonometric functions but are based on hyperbolas instead of circles. The function sinh(x) is defined as (e^x - e^(-x))/2. Understanding these functions is crucial for differentiating expressions involving them, as they have specific derivatives that differ from their trigonometric counterparts.

Chain Rule

The chain rule is a fundamental technique for finding the derivative of composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when differentiating functions like f(x) = sinh(4x), where the inner function is 4x.