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

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2:51 minutes

Problem 34

Textbook Question

Calculate the derivative of the following functions.
y = csc e^x

Verified step by step guidance

Step 1: Identify the function to differentiate. The function given is \( y = \csc(e^x) \).

Step 2: Recall the derivative of the cosecant function. The derivative of \( \csc(u) \) with respect to \( u \) is \( -\csc(u)\cot(u) \).

Step 3: Apply the chain rule. Since \( u = e^x \), differentiate \( u \) with respect to \( x \), which is \( \frac{du}{dx} = e^x \).

Step 4: Combine the results using the chain rule. The derivative of \( y = \csc(e^x) \) is \( \frac{dy}{dx} = -\csc(e^x)\cot(e^x) \cdot e^x \).

Step 5: Simplify the expression if possible. The derivative is \( \frac{dy}{dx} = -e^x \csc(e^x)\cot(e^x) \).

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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 of the function at any given point. The derivative is often denoted as f'(x) or dy/dx, and it can be calculated using various rules such as the power rule, product rule, and chain rule.

Cosecant Function (csc)

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). In the context of derivatives, understanding the properties and behavior of trigonometric functions like cosecant is essential, especially when applying differentiation rules. The derivative of csc(x) is -csc(x)cot(x), which is crucial for differentiating functions involving csc.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate 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 important when dealing with functions like y = csc(e^x), where e^x is the inner function.

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