Chain Rule examples AP Calculus AB

🧠 Quick Study Guide: Mastering the Chain Rule

• 🎯 What it is: The Chain Rule is a fundamental differentiation rule used to find the derivative of a composite function. A composite function is essentially a 'function within a function', like $f(g(x))$.
• 📝 The Formula: If $y = f(u)$ and $u = g(x)$, then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
• 🔢 In terms of functions: If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$. This means you take the derivative of the 'outer' function, keeping the 'inner' function the same, and then multiply by the derivative of the 'inner' function.
• ✨ Mnemonic Device: Think of it as 'Derivative of the outside, leave the inside alone, times derivative of the inside.'
• 🧮 Common Forms:
  • 🔗 Power Rule with Chain Rule: For $y = (g(x))^n$, then $\frac{dy}{dx} = n(g(x))^{n-1} \cdot g'(x)$.
  • 📈 Trigonometric Functions: For example, if $y = \sin(g(x))$, then $\frac{dy}{dx} = \cos(g(x)) \cdot g'(x)$. Similarly for other trig functions.
  • Exponential Functions: For $y = e^{g(x)}$, then $\frac{dy}{dx} = e^{g(x)} \cdot g'(x)$.
  • Logarithmic Functions: For $y = \ln(g(x))$, then $\frac{dy}{dx} = \frac{1}{g(x)} \cdot g'(x)$.
• ⚠️ Key Tip: Always identify the 'inner' and 'outer' functions clearly before applying the rule. This is crucial for avoiding common errors.

💡 Practice Quiz: Chain Rule Examples

1. What is the derivative of $f(x) = (3x^2 - 5x + 1)^4$?
   1. $4(3x^2 - 5x + 1)^3$
   2. $4(6x - 5)$
   3. $4(3x^2 - 5x + 1)^3 (6x - 5)$
   4. $(3x^2 - 5x + 1)^3 (6x - 5)$
2. Find $\frac{dy}{dx}$ if $y = \sin(4x^3 - 2x)$.
   1. $\cos(4x^3 - 2x)$
   2. $(12x^2 - 2)\cos(4x^3 - 2x)$
   3. $-\cos(4x^3 - 2x)(12x^2 - 2)$
   4. $(4x^3 - 2x)\cos(4x^3 - 2x)$
3. Differentiate $g(x) = e^{5x^2 - 3}$.
   1. $e^{5x^2 - 3}$
   2. $10x e^{5x^2 - 3}$
   3. $(5x^2 - 3)e^{5x^2 - 3}$
   4. $e^{10x}$
4. If $h(t) = \sqrt{t^2 + 9}$, what is $h'(t)$?
   1. $\frac{1}{2\sqrt{t^2 + 9}}$
   2. $\frac{t}{\sqrt{t^2 + 9}}$
   3. $2t\sqrt{t^2 + 9}$
   4. $\frac{2t}{\sqrt{t^2 + 9}}$
5. Calculate the derivative of $y = \tan(\frac{x}{2})$.
   1. $\sec^2(\frac{x}{2})$
   2. $\frac{1}{2}\sec^2(\frac{x}{2})$
   3. $2\sec^2(\frac{x}{2})$
   4. $\sec(\frac{x}{2})\tan(\frac{x}{2})$
6. Find the derivative of $f(x) = \ln(x^4 + 7x)$.
   1. $\frac{1}{x^4 + 7x}$
   2. $\frac{4x^3 + 7}{x^4 + 7x}$
   3. $(x^4 + 7x)(4x^3 + 7)$
   4. $\frac{1}{4x^3 + 7}$
7. What is the derivative of $y = (\cos x)^3$?
   1. $3\cos^2 x$
   2. $-3\sin x\cos^2 x$
   3. $3\sin x\cos^2 x$
   4. $-3\cos^2 x$