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Differentiation of Composite & Inverse Functions

Emily Davis

Emily Davis

5 min read

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Study Guide Overview

This study guide covers second derivatives and higher-order derivatives. It explains how to find these derivatives, their notation (f(x)f''(x), f(3)(x)f^{(3)}(x), racd2ydx2rac{d^2y}{dx^2}, etc.), and their applications, such as analyzing concavity and using L'Hopital's Rule. It includes worked examples, practice questions, and a glossary of key terms like jerk.

Table of Contents

  1. Second Derivatives
  2. Higher-Order Derivatives
  3. Practice Questions
  4. Glossary
  5. Summary and Key Takeaways

Second Derivatives

What is a second derivative?

The second derivative of a function is the derivative of the derivative. It measures the rate of change of the rate of change of a function.

Key Points:

  • The second derivative is denoted as:
    • f(x)f''(x)
    • d2ydx2\frac{d^2 y}{dx^2}
    • yy''
  • The second derivative helps in understanding the concavity of a function's graph.

To find the second derivative, differentiate the function and then differentiate the result once more. Both the function and its first derivative must be differentiable for this to work.

Exam Tip

Remember that the second derivative can provide insights into the concavity and inflection points of a function's graph.

Worked Example

A function ff is defined by f(x)=3x32x23x+2f(x) = 3x^3 - 2x^2 - 3x + 2. Find the second derivative of f(x)f(x).

Solution:

First, find the first derivative: f(x)=9x24x3f'(x) = 9x^2 - 4x - 3

Then, find the second derivative: f(x)=18x4f''(x) = 18x - 4


Higher-Order Derivatives

What are higher-order derivatives?

Higher-order derivatives extend the concept of the second derivative by continuing to differentiate the function multiple times, provided the derivatives remain differentiable.

Key Points:

  • Higher-order derivatives are denoted as:
    • f(x),f(x),f(3)(x),f(4)(x)f'(x), f''(x), f^{(3)}(x), f^{(4)}(x)
    • dydx,d2ydx2,d3ydx3,d4ydx4\frac{dy}{dx}, \frac{d^2 y}{dx^2}, \frac{d^3 y}{dx^3}, \frac{d^4 y}{dx^4}
    • y,y,y(3),y(4)y', y'', y^{(3)}, y^{(4)}
  • The third derivative measures the rate of change of the rate of change of the rate of change of a function.

In motion, if a function describes the displacement of an object:

  • The first derivative represents velocity.
  • The second derivative represents acceleration.
  • The third derivative represents the rate of change of acceleration (jerk).

Application: Higher-order derivatives are useful in applying L'Hopital's Rule for evaluating limits: limxaf(x)g(x)=limxaf(x)g(x)=limxaf(x)g(x)=\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} = \lim_{x \to a} \frac{f''(x)}{g''(x)} = \ldots

Worked Example

(a) Find the third derivative of f(x)=4x33x2+9x8f(x) = 4x^3 - 3x^2 + 9x - 8.

Solution:

First, find the first derivative: f(x)=12x26x+9f'(x) = 12x^2 - 6x + 9

Next, find the second derivative: f(x)=24x6f''(x) = 24x - 6

Finally, find the third derivative: f(3)(x)=24f^{(3)}(x) = 24

(b) Given that y=sinxy = \sin x, find d4ydx4\frac{d^4 y}{dx^4}.

Solution:

First, find the first derivative: dydx=cosx\frac{dy}{dx} = \cos x

Next, find the second derivative: d2ydx2=sinx\frac{d^2 y}{dx^2} = -\sin x

Then, find the third derivative: d3ydx3=cosx\frac{d^3 y}{dx^3} = -\cos x

Finally, find the fourth derivative: d4ydx4=sinx\frac{d^4 y}{dx^4} = \sin x


Practice Questions

Practice Question
  1. Find the second derivative of g(x)=2x45x3+x7g(x) = 2x^4 - 5x^3 + x - 7.
Practice Question
  1. If h(t)=cos(t)h(t) = \cos(t), determine d3hdt3\frac{d^3 h}{dt^3}.
Practice Question
  1. Given k(x)=exsin(x)k(x) = e^x \cdot \sin(x), find the second derivative of k(x)k(x).

Glossary

  • Second Derivative: The derivative of the derivative of a function.
  • Higher-Order Derivative: Derivatives beyond the second derivative, such as third, fourth, etc.
  • Concavity: Describes whether a function is curving upwards or downwards.
  • L'Hopital's Rule: A method to evaluate limits involving indeterminate forms.
  • Jerk: The rate of change of acceleration.

Summary and Key Takeaways

Summary

  • The second derivative is the derivative of the first derivative and provides information about the concavity of a function's graph.
  • Higher-order derivatives extend this concept to third, fourth, and further derivatives, each measuring the rate of change of the previous derivative.
  • They are particularly useful in physics for describing motion and in calculus for applying L'Hopital's Rule.

Key Takeaways

  1. Second Derivative: Denoted by f(x)f''(x) or d2ydx2\frac{d^2 y}{dx^2}, measures how the rate of change is changing.
  2. Higher-Order Derivatives: Third derivative f(3)(x)f^{(3)}(x), fourth derivative f(4)(x)f^{(4)}(x), etc.
  3. Concavity: Determines the shape of the graph.
  4. Applications: Motion analysis and L'Hopital's Rule.
Exam Tip

Always check the differentiability of the function and its derivatives before proceeding with higher-order derivatives.


Exam Strategy

  • Focus on understanding the physical interpretation of derivatives, especially in motion problems.
  • Practice differentiating functions multiple times to become comfortable with higher-order derivatives.
  • Use L'Hopital's Rule strategically to simplify limit problems.
Common Mistake

Do not confuse the notation for higher-order derivatives, especially when dealing with trigonometric and exponential functions.

Review the related topic on 'Concavity of Functions' for a deeper understanding of how second derivatives affect graph shapes.

Question 1 of 10

Ready to flex your calculus muscles? 💪 What is the second derivative of f(x)=5x2+3x7f(x) = 5x^2 + 3x - 7?

10x+310x + 3

5x25x^2

1010

00