AP Calculus AB/BC: Derivatives — Rules, Chain Rule, Implicit Differentiation

TL;DR

Derivatives tell you how a function's output changes relative to its input, representing instantaneous rates of change or slopes of tangent lines. You'll use basic rules for common functions, apply the Chain Rule for composite functions, and use implicit differentiation when 'y' isn't explicitly defined. Mastering these skills is crucial for understanding function behavior and solving calculus problems.

1. The Mental Model

Think of differentiation as finding the "speedometer reading" of a function at any given point. It tells you exactly how fast one quantity is changing with respect to another right at that moment.

2. The Core Material

When you're finding the derivative, you're essentially looking for a new function that describes the slope of the original function's tangent line at every point.

Basic Derivative Rules

These are your building blocks. You'll use these rules constantly.

• Constant Rule: If $f(x) = c$ (where $c$ is any number), then $f'(x) = 0$. The slope of a horizontal line is always zero.
• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Just bring the exponent down as a coefficient and subtract 1 from the exponent.
  • Example: If $f(x) = x^4$, then $f'(x) = 4x^3$.
  • Example: If $f(x) = \frac{1}{x} = x^{-1}$, then $f'(x) = -1x^{-2} = -\frac{1}{x^2}$.
• Constant Multiple Rule: If $g(x) = c \cdot f(x)$, then $g'(x) = c \cdot f'(x)$. You can pull constants out.
  • Example: If $f(x) = 5x^3$, then $f'(x) = 5 \cdot (3x^2) = 15x^2$.
• Sum/Difference Rule: If $h(x) = f(x) \pm g(x)$, then $h'(x) = f'(x) \pm g'(x)$. You can differentiate term by term.
  • Example: If $f(x) = 3x^2 + 2x - 7$, then $f'(x) = 6x + 2 - 0 = 6x + 2$.
• Product Rule: If $h(x) = f(x) \cdot g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.
  • Mnemonic: first D second + second D first.
• Quotient Rule: If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.
  • Mnemonic: low D high minus high D low, all over low squared.

Derivative of Common Functions

You'll need to memorize these:

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$
• $\frac{d}{dx}(\sec x) = \sec x \tan x$
• $\frac{d}{dx}(\csc x) = -\csc x \cot x$
• $\frac{d}{dx}(\cot x) = -\csc^2 x$
• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(a^x) = a^x \ln a$
• $\frac{d}{dx}(\ln x) = \frac{1}{x}$
• $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$
• $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$

The Chain Rule

This is for "function composition" – when one function is inside another, like $f(g(x))$. Think of it like peeling an onion, working from the outside in. You differentiate the outer function, leaving the inner function untouched, then multiply by the derivative of the inner function.

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

graph TD
    A["Start with Composite Function"] --> B["Identify Outer Function f()"];
    B --> C["Identify Inner Function g()"];
    C --> D["Differentiate Outer Function f'(...)"];
    D --> E["Keep Inner Function Same (g(x))"];
    E --> F["Multiply by Derivative of Inner Function (g'(x))"];
    F --> G["Combine: f'(g(x)) * g'(x)"];

• Example: If $y = \sin(x^2)$. Here, $f(u) = \sin u$ and $u = g(x) = x^2$.
  • $f'(u) = \cos u$
  • $g'(x) = 2x$
  • So, $y' = \cos(x^2) \cdot 2x = 2x \cos(x^2)$.

Implicit Differentiation

You use this when 'y' isn't explicitly defined as a function of 'x' (e.g., $y = x^2$). Instead, 'y' is mixed in an equation with 'x' (e.g., $x^2 + y^2 = 25$). The key is remembering that $y$ is some function of $x$, so when you differentiate a term with $y$ in it, you'll need the Chain Rule and attach a $\frac{dy}{dx}$.

1. Differentiate both sides of the equation with respect to 'x'.
2. Any term involving 'x' is differentiated normally.
3. Any term involving 'y' is differentiated normally and then multiplied by $\frac{dy}{dx}$ (due to the Chain Rule, as $y$ is a function of $x$).
4. Isolate $\frac{dy}{dx}$ algebraically.

• Example: Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$.
  • Differentiate $x^2$ wrt x: $2x$.
  • Differentiate $y^2$ wrt x: $2y \frac{dy}{dx}$ (using Power Rule + Chain Rule).
  • Differentiate $25$ wrt x: $0$.
  • So, $2x + 2y \frac{dy}{dx} = 0$.
  • Solve for $\frac{dy}{dx}$: $2y \frac{dy}{dx} = -2x \implies \frac{dy}{dx} = -\frac{2x}{2y} = -\frac{x}{y}$.

3. Worked Example

Let's find the derivative $\frac{dy}{dx}$ of the equation $x^2y + \sin(y^2) = 3x$. This involves product rule, chain rule, and implicit differentiation.

1. Differentiate each term with respect to $x$:

   • For $x^2y$: This is a product, so use the Product Rule $f'(x)g(x) + f(x)g'(x)$.
     • Let $f(x) = x^2 \Rightarrow f'(x) = 2x$.
     • Let $g(x) = y \Rightarrow g'(x) = \frac{dy}{dx}$ (since $y$ is a function of $x$).
     • So, $\frac{d}{dx}(x^2y) = (2x)y + x^2\left(\frac{dy}{dx}\right)$.
   • For $\sin(y^2)$: This requires the Chain Rule, as $y^2$ is inside $\sin()$.
     • Outer function is $\sin(u)$, derivative is $\cos(u)$. Inner function is $y^2$, derivative is $2y\frac{dy}{dx}$.
     • So, $\frac{d}{dx}(\sin(y^2)) = \cos(y^2) \cdot \left(2y\frac{dy}{dx}\right)$.
   • For $3x$: Differentiate normally.
     • So, $\frac{d}{dx}(3x) = 3$.

2. Put it all together:
   $2xy + x^2\frac{dy}{dx} + 2y\cos(y^2)\frac{dy}{dx} = 3$

3. Isolate terms with $\frac{dy}{dx}$:
   $x^2\frac{dy}{dx} + 2y\cos(y^2)\frac{dy}{dx} = 3 - 2xy$

4. Factor out $\frac{dy}{dx}$:
   $\frac{dy}{dx}(x^2 + 2y\cos(y^2)) = 3 - 2xy$

5. Solve for $\frac{dy}{dx}$:
   $\frac{dy}{dx} = \frac{3 - 2xy}{x^2 + 2y\cos(y^2)}$

4. Key Takeaways

• Always identify the outer function first when applying the Chain Rule.
• When differentiating implicitly, remember to multiply by $\frac{dy}{dx}$ every time you differentiate a term containing $y$.
• The Product Rule ($uv' + vu'$) and Quotient Rule ($\frac{vu' - uv'}{v^2}$) are specific patterns for differentiating products and quotients.
• Memorize the derivatives of common functions (trig, exponential, log, inverse trig).
• Derivative rules can often be combined in complex problems; break them down step-by-step.

Common Mistakes to Avoid

• Forgetting the Chain Rule entirely when one function is inside another.
• Applying the Product or Quotient Rule unnecessarily, especially with constants (e.g., $d/dx(5x^3)$ is $15x^2$, not requiring the Product Rule).
• In implicit differentiation, forgetting to differentiate the constant term on the right side of the equation to zero.
• Algebraic errors when isolating $\frac{dy}{dx}$ after implicit differentiation.
• Confusing the derivatives of $\tan x$ ($\sec^2 x$) and $\sec x$ ($\sec