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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2:34 minutes

Problem 37

Textbook Question

Calculate the derivative of the following functions.
y = sin (4x³ + 3x +1)

Verified step by step guidance

Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sin(u) \) and the inner function is \( u = 4x^3 + 3x + 1 \).

Step 2: Apply the chain rule for differentiation, which states that \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Step 3: Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \).

Step 4: Differentiate the inner function \( u = 4x^3 + 3x + 1 \) with respect to \( x \). The derivative is \( 12x^2 + 3 \).

Step 5: Combine the results from steps 3 and 4 using the chain rule: \( \frac{dy}{dx} = \cos(4x^3 + 3x + 1) \cdot (12x^2 + 3) \).

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

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 y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions that are nested within each other, such as the sine function in the given problem.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to the ratios of sides in right triangles. In calculus, these functions are important because they have specific derivatives: for example, the derivative of sin(x) is cos(x). Understanding how to differentiate these functions is crucial when working with problems involving trigonometric expressions, as seen in the given function.

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