Differentiation of Composite & Inverse Functions

#Table of Contents

1. Second Derivatives
   • What is a second derivative?
   • Worked Example
2. Higher-Order Derivatives
   • What are higher-order derivatives?
   • Worked Example
3. Practice Questions
4. Glossary
5. Summary and Key Takeaways

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#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:
  • $$f''(x)$$
  • $$\frac{d^2 y}{dx^2}$$
  • $$y''$$
• 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.

#Worked Example

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

Solution:

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

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

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#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:
  • $$f'(x), f''(x), f^{(3)}(x), f^{(4)}(x)$$
  • $$\frac{dy}{dx}, \frac{d^2 y}{dx^2}, \frac{d^3 y}{dx^3}, \frac{d^4 y}{dx^4}$$
  • $$y', y'', y^{(3)}, y^{(4)}$$
• 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: $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} = \lim_{x \to a} \frac{f''(x)}{g''(x)} = \ldots$$

#Worked Example

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

Solution:

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

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

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

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

Solution:

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

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

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

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

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#Practice Questions

Practice Question

1. Find the second derivative of $g(x) = 2x^4 - 5x^3 + x - 7$.

Practice Question

2. If $h(t) = \cos(t)$, determine $\frac{d^3 h}{dt^3}$.

Practice Question

3. Given $k(x) = e^x \cdot \sin(x)$, find the second derivative of $k(x)$.

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#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.

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

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#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.

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