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

Higher Order Derivatives

Calculus

3. Techniques of Differentiation

Higher Order Derivatives

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3:05 minutes

Problem 3.3.4

Textbook Question

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

w = 3z⁷ − 7z³ + 21z²

Verified step by step guidance

Step 1: Identify the function for which you need to find the derivatives. Here, the function is \( w = 3z^7 - 7z^3 + 21z^2 \).

Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that the derivative of \( z^n \) is \( nz^{n-1} \).

Step 3: Calculate the first derivative \( \frac{dw}{dz} \) by applying the power rule: \( \frac{d}{dz}(3z^7) = 21z^6 \), \( \frac{d}{dz}(-7z^3) = -21z^2 \), and \( \frac{d}{dz}(21z^2) = 42z \). Combine these results to get the first derivative.

Step 4: To find the second derivative, apply the power rule again to the first derivative. Differentiate each term of the first derivative.

Step 5: Calculate the second derivative \( \frac{d^2w}{dz^2} \) by applying the power rule to each term of the first derivative obtained in Step 3. Combine these results to get the second derivative.

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Key Concepts

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

Derivative

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The first derivative of a function provides information about its slope, while the second derivative gives insight into the function's concavity and acceleration.

Power Rule

The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of ax^n, where a is a constant and n is a real number. According to this rule, the derivative is calculated by multiplying the coefficient by the exponent and then reducing the exponent by one. This rule simplifies the process of finding derivatives for polynomial functions.

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative gives the rate of change, the second derivative provides information about the curvature of the graph, and further derivatives can indicate more complex behaviors of the function. Understanding higher-order derivatives is essential for analyzing the behavior of functions in greater detail.

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