Core Differentiation Rules
Product Rule
Formula: $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$
In words: The derivative of a product equals the first function times the derivative of the second function, plus the second function times the derivative of the first function.
Example: $\frac{d}{dx}[x^2 \cdot \sin(x)]$
- $f(x) = x^2$, $f'(x) = 2x$
- $g(x) = \sin(x)$, $g'(x) = \cos(x)$
- $\frac{d}{dx}[x^2 \cdot \sin(x)] = 2x \cdot \sin(x) + x^2 \cdot \cos(x)$
Quotient Rule
Formula: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) – f(x) \cdot g'(x)}{[g(x)]^2}$
In words: The derivative of a quotient equals the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
Mnemonic: “Low d-high minus high d-low, all over low squared”
Example: $\frac{d}{dx}\left[\frac{x^2}{\cos(x)}\right]$
- $f(x) = x^2$, $f'(x) = 2x$
- $g(x) = \cos(x)$, $g'(x) = -\sin(x)$
- $\frac{d}{dx}\left[\frac{x^2}{\cos(x)}\right] = \frac{2x \cdot \cos(x) – x^2 \cdot (-\sin(x))}{[\cos(x)]^2} = \frac{2x\cos(x) + x^2\sin(x)}{\cos^2(x)}$
Chain Rule
Formula: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
In words: The derivative of a composite function equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Example: $\frac{d}{dx}[\sin(x^2)]$
- $f(u) = \sin(u)$, $f'(u) = \cos(u)$
- $g(x) = x^2$, $g'(x) = 2x$
- $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)$
Product Rule Applications
1. Product of Two Functions
Formula: $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$
Example: $\frac{d}{dx}[e^x \cdot \ln(x)]$
- $f(x) = e^x$, $f'(x) = e^x$
- $g(x) = \ln(x)$, $g'(x) = \frac{1}{x}$
- $\frac{d}{dx}[e^x \cdot \ln(x)] = e^x \cdot \ln(x) + e^x \cdot \frac{1}{x} = e^x\ln(x) + \frac{e^x}{x}$
2. Product of Three Functions
Formula: $\frac{d}{dx}[f(x) \cdot g(x) \cdot h(x)]$ $= f'(x) \cdot g(x) \cdot h(x) + f(x) \cdot g'(x) \cdot h(x) + f(x) \cdot g(x) \cdot h'(x)$
Example: $\frac{d}{dx}[x \cdot \sin(x) \cdot e^x]$
- $f(x) = x$, $f'(x) = 1$
- $g(x) = \sin(x)$, $g'(x) = \cos(x)$
- $h(x) = e^x$, $h'(x) = e^x$
- $\frac{d}{dx}[x \cdot \sin(x) \cdot e^x] = 1 \cdot \sin(x) \cdot e^x + x \cdot \cos(x) \cdot e^x + x \cdot \sin(x) \cdot e^x$
- $= e^x\sin(x) + xe^x\cos(x) + xe^x\sin(x) = e^x\sin(x)(1 + x) + xe^x\cos(x)$
3. Product with Constants
Formula: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$
Example: $\frac{d}{dx}[5 \cdot x^3]$
- $f(x) = x^3$, $f'(x) = 3x^2$
- $\frac{d}{dx}[5 \cdot x^3] = 5 \cdot 3x^2 = 15x^2$
4. Product Rule with Trigonometric Functions
Formula: $\frac{d}{dx}[\sin(x) \cdot \cos(x)] = \cos^2(x) – \sin^2(x)$
Example:
- $f(x) = \sin(x)$, $f'(x) = \cos(x)$
- $g(x) = \cos(x)$, $g'(x) = -\sin(x)$
- $\frac{d}{dx}[\sin(x) \cdot \cos(x)] = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x))$
- $= \cos^2(x) – \sin^2(x) = \cos(2x)$
Quotient Rule Applications
1. Basic Quotient Rule
Formula: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) – f(x) \cdot g'(x)}{[g(x)]^2}$
Example: $\frac{d}{dx}\left[\frac{x^2}{e^x}\right]$
- $f(x) = x^2$, $f'(x) = 2x$
- $g(x) = e^x$, $g'(x) = e^x$
- $\frac{d}{dx}\left[\frac{x^2}{e^x}\right] = \frac{2x \cdot e^x – x^2 \cdot e^x}{(e^x)^2} = \frac{2x – x^2}{e^x}$
2. Quotient with Trigonometric Functions
Formula: $\frac{d}{dx}\left[\frac{\sin(x)}{\cos(x)}\right] = \frac{d}{dx}[\tan(x)] = \sec^2(x)$
Example:
- $f(x) = \sin(x)$, $f'(x) = \cos(x)$
- $g(x) = \cos(x)$, $g'(x) = -\sin(x)$
- $\frac{d}{dx}\left[\frac{\sin(x)}{\cos(x)}\right] = \frac{\cos(x) \cdot \cos(x) – \sin(x) \cdot (-\sin(x))}{[\cos(x)]^2}$
- $= \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)} = \sec^2(x)$
3. Quotient with Logarithmic and Polynomial Functions
Formula: $\frac{d}{dx}\left[\frac{\ln(x)}{x^n}\right]$
Example: $\frac{d}{dx}\left[\frac{\ln(x)}{x^2}\right]$
- $f(x) = \ln(x)$, $f'(x) = \frac{1}{x}$
- $g(x) = x^2$, $g'(x) = 2x$
- $\frac{d}{dx}\left[\frac{\ln(x)}{x^2}\right] = \frac{\frac{1}{x} \cdot x^2 – \ln(x) \cdot 2x}{(x^2)^2} = \frac{x – 2x\ln(x)}{x^4} = \frac{1 – 2\ln(x)}{x^3}$
4. Reciprocal Rule
Formula: $\frac{d}{dx}\left[\frac{1}{f(x)}\right] = -\frac{f'(x)}{[f(x)]^2}$
Example: $\frac{d}{dx}\left[\frac{1}{\sin(x)}\right] = \frac{d}{dx}[\csc(x)]$
- $f(x) = \sin(x)$, $f'(x) = \cos(x)$
- $\frac{d}{dx}\left[\frac{1}{\sin(x)}\right] = -\frac{\cos(x)}{[\sin(x)]^2} = -\frac{\cos(x)}{\sin^2(x)} = -\cot(x)\csc(x)$
Chain Rule Applications
1. Basic Chain Rule
Formula: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
Example: $\frac{d}{dx}[e^{3x}]$
- $f(u) = e^u$, $f'(u) = e^u$
- $g(x) = 3x$, $g'(x) = 3$
- $\frac{d}{dx}[e^{3x}] = e^{3x} \cdot 3 = 3e^{3x}$
2. Chain Rule with Power Functions
Formula: $\frac{d}{dx}[g(x)^n] = n \cdot g(x)^{n-1} \cdot g'(x)$
Example: $\frac{d}{dx}[(2x+1)^3]$
- $f(u) = u^3$, $f'(u) = 3u^2$
- $g(x) = 2x+1$, $g'(x) = 2$
- $\frac{d}{dx}[(2x+1)^3] = 3(2x+1)^2 \cdot 2 = 6(2x+1)^2$
3. Chain Rule with Trigonometric Functions
Formula: $\frac{d}{dx}[\sin(g(x))] = \cos(g(x)) \cdot g'(x)$
Example: $\frac{d}{dx}[\sin(x^2)]$
- $f(u) = \sin(u)$, $f'(u) = \cos(u)$
- $g(x) = x^2$, $g'(x) = 2x$
- $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)$
4. Multiple Chain Rule (Nested Functions)
Formula: $\frac{d}{dx}[f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$
Example: $\frac{d}{dx}[\ln(\sin(x^2))]$
- $f(u) = \ln(u)$, $f'(u) = \frac{1}{u}$
- $g(v) = \sin(v)$, $g'(v) = \cos(v)$
- $h(x) = x^2$, $h'(x) = 2x$
- $\frac{d}{dx}[\ln(\sin(x^2))] = \frac{1}{\sin(x^2)} \cdot \cos(x^2) \cdot 2x = \frac{2x\cos(x^2)}{\sin(x^2)} = 2x\cot(x^2)$
Combined Rules and Special Cases
1. Product and Chain Rules Together
Formula: $\frac{d}{dx}[f(x) \cdot g(h(x))]$ $= f'(x) \cdot g(h(x)) + f(x) \cdot g'(h(x)) \cdot h'(x)$
Example: $\frac{d}{dx}[x^2 \cdot \sin(3x)]$
- $f(x) = x^2$, $f'(x) = 2x$
- $g(h(x)) = \sin(3x)$, $g'(h(x)) = \cos(3x)$, $h'(x) = 3$
- $\frac{d}{dx}[x^2 \cdot \sin(3x)] = 2x \cdot \sin(3x) + x^2 \cdot \cos(3x) \cdot 3 = 2x\sin(3x) + 3x^2\cos(3x)$
2. Quotient and Chain Rules Together
Formula: $\frac{d}{dx}\left[\frac{f(x)}{g(h(x))}\right]$ $= \frac{f'(x) \cdot g(h(x)) – f(x) \cdot g'(h(x)) \cdot h'(x)}{[g(h(x))]^2}$
Example: $\frac{d}{dx}\left[\frac{x^3}{\cos(2x)}\right]$
- $f(x) = x^3$, $f'(x) = 3x^2$
- $g(h(x)) = \cos(2x)$, $g'(h(x)) = -\sin(2x)$, $h'(x) = 2$
- $\frac{d}{dx}\left[\frac{x^3}{\cos(2x)}\right] = \frac{3x^2 \cdot \cos(2x) – x^3 \cdot (-\sin(2x)) \cdot 2}{[\cos(2x)]^2}$
- $= \frac{3x^2\cos(2x) + 2x^3\sin(2x)}{\cos^2(2x)}$
3. Chain Rule with Exponential Functions
Formula: $\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)$
Example: $\frac{d}{dx}[e^{\sin(x)}]$
- $f(u) = e^u$, $f'(u) = e^u$
- $g(x) = \sin(x)$, $g'(x) = \cos(x)$
- $\frac{d}{dx}[e^{\sin(x)}] = e^{\sin(x)} \cdot \cos(x)$
4. Chain Rule with Logarithmic Functions
Formula: $\frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}$
Example: $\frac{d}{dx}[\ln(x^2 + 1)]$
- $f(u) = \ln(u)$, $f'(u) = \frac{1}{u}$
- $g(x) = x^2 + 1$, $g'(x) = 2x$
- $\frac{d}{dx}[\ln(x^2 + 1)] = \frac{2x}{x^2 + 1}$
Implicit Differentiation
Using the Chain Rule for Implicit Differentiation
Process: Differentiate both sides with respect to x, treating y as a function of x.
Example: Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$
- Differentiate both sides: $\frac{d}{dx}[x^2 + y^2] = \frac{d}{dx}[25]$
- Apply chain rule to $y^2$: $2x + 2y \cdot \frac{dy}{dx} = 0$
- Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$
Quick Reference: Product, Quotient and Chain Rules
Product Rule Shortcuts
- $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$
- $\frac{d}{dx}[f(x) \cdot g(x) \cdot h(x)] = f’g \cdot h + f \cdot g’h + f \cdot g \cdot h’$
- $\frac{d}{dx}[x^n \cdot f(x)] = nx^{n-1} \cdot f(x) + x^n \cdot f'(x)$
Quotient Rule Shortcuts
- $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) – f(x) \cdot g'(x)}{[g(x)]^2}$
- $\frac{d}{dx}\left[\frac{1}{f(x)}\right] = -\frac{f'(x)}{[f(x)]^2}$
- $\frac{d}{dx}\left[\frac{f(x)}{x^n}\right] = \frac{f'(x) \cdot x^n – f(x) \cdot nx^{n-1}}{x^{2n}} = \frac{x \cdot f'(x) – n \cdot f(x)}{x^{n+1}}$
Chain Rule Shortcuts
- $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
- $\frac{d}{dx}[(g(x))^n] = n \cdot (g(x))^{n-1} \cdot g'(x)$
- $\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)$
- $\frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}$
- $\frac{d}{dx}[\sin(g(x))] = \cos(g(x)) \cdot g'(x)$
- $\frac{d}{dx}[\cos(g(x))] = -\sin(g(x)) \cdot g'(x)$
Common Mistakes and How to Avoid Them
Product Rule Mistakes
- Mistake: Multiplying the derivatives instead of following the rule
- Correct approach: Remember the formula $f’g + fg’$, not $f’g’$
Quotient Rule Mistakes
- Mistake: Getting the numerator order wrong
- Correct approach: Use the mnemonic “Low d-high minus high d-low”
- Mistake: Forgetting to square the denominator
- Correct approach: Always divide by $[g(x)]^2$
Chain Rule Mistakes
- Mistake: Forgetting to evaluate the outer function at the inner function
- Correct approach: Always write $f'(g(x))$, not just $f'(x)$
- Mistake: Forgetting to multiply by the derivative of the inner function
- Correct approach: Always multiply by $g'(x)$
Practice Strategies
- Break down complex expressions into simpler parts before differentiating
- Identify the structure of the expression (product, quotient, or composition)
- Label the functions clearly (f(x), g(x), etc.)
- Apply the appropriate rule methodically
- Check your work by comparing with known derivatives or verifying with the definition
This comprehensive cheatsheet covers the key differentiation rules (Product, Quotient, and Chain rules) in calculus, providing clear formulas, examples, and applications to help you master these essential calculus techniques.