#AP Calculus AB/BC: Higher-Order Derivatives - Your Ultimate Review 🚀

Hey there, future AP Calculus master! Let's tackle higher-order derivatives (HODs) together. This guide is designed to be your go-to resource, especially the night before the exam. We'll make sure everything clicks, and you'll feel confident and ready!

#Understanding Higher-Order Derivatives

#What are Higher-Order Derivatives?

Don't let the name intimidate you! Higher-order derivatives are simply the derivatives of derivatives. We can find the first derivative, second derivative, third derivative, and so on. Each one gives us unique insights into the function's behavior. 💡

• First Derivative (f'(x)): Tells us about the function's slope, where the function has relative minima or maxima (where f'(x) = 0).
• Second Derivative (f''(x)): Reveals the concavity of the function and helps us find inflection points (where the concavity changes).

#Notation

Here's how we write them:

• Second Derivative: $f''(x)$, $y''$, $\frac{d^2y}{dx^2}$
• Higher-Order Derivatives: $f^n(x)$, $\frac{d^ny}{dx^n}$

#How to Calculate HODs

1. Find the first derivative f'(x).
2. Find the second derivative f''(x) by taking the derivative of f'(x).
3. Find the third derivative f'''(x) by taking the derivative of f''(x).
4. Continue this process for higher-order derivatives.

#Example Walkthrough

Let's calculate some HODs for $f(x) = \frac{2}{3}x^3 + 4x^2 + 3x - 1$:

1. First Derivative: Using the Power Rule, we get $f'(x) = 2x^2 + 8x + 3$.
2. Second Derivative: Applying the Power Rule again, we get $f''(x) = 4x + 8$.
3. Third Derivative: Differentiating again, we get $f'''(x) = 4$.
4. Fourth Derivative: Since the third derivative is a constant, $f^{(4)}(x) = 0$.

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#Practice Time! 🚀

Let's solidify your understanding with some practice problems.

#Set 1: Quick Power Rules and Trig

#Example 1: Power Rule

$f(x) = 6x^4 - 2x^2 + 5x + 1$

1. First Derivative: $f'(x) = 24x^3 - 4x + 5$
2. Second Derivative: $f''(x) = 72x^2 - 4$

#Example 2: Trig Functions

$f(x) = \sin(x)$

1. First Derivative: $f'(x) = \cos(x)$
2. Second Derivative: $f''(x) = -\sin(x)$

#Example 3: Composite Trig Functions (Chain Rule)

$f(x) = \cos(2x)$

1. First Derivative: Using the Chain Rule, $f'(x) = -2\sin(2x)$
2. Second Derivative: Using the Chain Rule again, $f''(x) = -4\cos(2x)$

#Set 2: Chain and Product Rule Galore

#Example 4: Chain and Product Rule

$f(x) = (5x^4 + 2x^2 - 3x + 9)^2$

1. First Derivative: Using the Chain Rule, $f'(x) = 2(5x^4 + 2x^2 - 3x + 9)(20x^3 + 4x - 3)$

2. Second Derivative: Using the Product Rule and Chain Rule, $f''(x) = (40x^3 + 8x - 6)(20x^3 + 4x - 3) + (10x^4 + 4x^2 - 6x + 18)(60x^2 + 4)$

   (Expanded: f''(x) = 1400x^6 + 600x^4 - 600x^3 + 1128x^2 - 72x + 90)

#Example 5: Chain Rule with Square Root

$f(x) = \sqrt{5x^3 + 81x^2} = (5x^3 + 81x^2)^{\frac{1}{2}}$

1. First Derivative: Using the Chain Rule, $f'(x) = \frac{1}{2}(5x^3 + 81x^2)^{-\frac{1}{2}}(15x^2 + 162x)$
2. Second Derivative: Using the Product and Chain Rules, $f''(x) = \frac{15(5x + 108)}{4(5x + 81)^{\frac{3}{2}}}$

#Set 3: Rational Functions and Natural Logs

#Example 6: Trig and Natural Log

$f(x) = \tan(3x) + \ln(x)$

1. First Derivative: $f'(x) = 3\sec^2(3x) + \frac{1}{x}$
2. Second Derivative: $f''(x) = 18\sec^2(3x)\tan(3x) - \frac{1}{x^2}$

#Example 7: Quotient Rule

$f(x) = \frac{x^2}{x + 1}$

1. First Derivative: Using the Quotient Rule, $f'(x) = \frac{2x(x + 1) - x^2}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2}$
2. Second Derivative: Using the Quotient Rule and Chain Rule, $f''(x) = \frac{2}{(x + 1)^3}$

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#Final Exam Focus

#Key Topics

• Power Rule, Chain Rule, Product Rule, Quotient Rule: Master these differentiation techniques.
• Trigonometric Derivatives: Know the derivatives of sine, cosine, and tangent.
• Natural Log Derivatives: Understand the derivative of ln(x).
• Composite Functions: Be comfortable using the Chain Rule.
• Applications of Derivatives: Understand how first and second derivatives relate to increasing/decreasing intervals, concavity, and inflection points.

#Common Question Types

• Multiple Choice: Expect quick calculations involving basic derivatives and applications.
• Free Response: Be prepared to show all steps for finding higher-order derivatives and applying them in context.

• Time Management: Don't spend too long on one question. Move on and come back if you have time.
• Show Your Work: Even if you make a small mistake, you can get partial credit for showing your process.
• Double-Check: Make sure you've applied the correct rules and haven't made simple arithmetic errors.
• Stay Calm: Take deep breaths and trust your preparation. You've got this!

#Practice Questions

Practice Question

#Multiple Choice Questions

1. If $f(x) = 3x^4 - 2x^3 + x - 5$, what is $f''(x)$? (A) $12x^3 - 6x^2 + 1$ (B) $36x^2 - 12x$ (C) $12x^2 - 6x$ (D) $36x^2 - 12x + 1$

2. If $f(x) = sin(2x)$, what is $f''(x)$? (A) $2cos(2x)$ (B) $-4sin(2x)$ (C) $-2sin(2x)$ (D) $4cos(2x)$

3. If $f(x) = ln(x^2)$, what is $f''(x)$? (A) $\frac{2}{x}$ (B) $-\frac{2}{x^2}$ (C) $\frac{1}{x^2}$ (D) $-\frac{1}{x^2}$

#Free Response Question

Consider the function $f(x) = x^3 - 6x^2 + 9x + 1$.

(a) Find $f'(x)$ and $f''(x)$. (2 points)

(b) Determine the intervals where $f(x)$ is increasing and decreasing. (3 points)

(c) Find the x-coordinates of all inflection points of $f(x)$. (2 points)

(d) Determine the intervals where $f(x)$ is concave up and concave down. (2 points)

Scoring Breakdown: (a) - 1 point for correctly finding $f'(x) = 3x^2 - 12x + 9$ - 1 point for correctly finding $f''(x) = 6x - 12$

(b) - 1 point for setting $f'(x) = 0$ and finding critical points $x = 1, 3$ - 1 point for creating a sign chart for $f'(x)$ - 1 point for stating intervals: increasing on $(-\infty, 1)$ and $(3, \infty)$, decreasing on $(1, 3)$

(c) - 1 point for setting $f''(x) = 0$ and finding $x = 2$ - 1 point for confirming that concavity changes at $x = 2$

(d) - 1 point for creating a sign chart for $f''(x)$ - 1 point for stating intervals: concave down on $(-\infty, 2)$, concave up on $(2, \infty)$

You've got this! Let's ace that exam! 🎉