Derivatives and Differentiation
Grade 12 · Math · Study Guide
A comprehensive review of derivative rules, techniques, and their geometric interpretations for advanced calculus students.
A derivative measures the instantaneous rate of change of a function at a given point. Geometrically, the derivative f'(x) represents the slope of the tangent line to the curve y = f(x) at point x. The formal definition uses the limit: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.
Key differentiation rules make computing derivatives efficient. The Power Rule states d/dx[x^n] = nx^(n-1). The Product Rule states d/dx[f·g] = f'g + fg'. The Quotient Rule states d/dx[f/g] = (f'g - fg') / g². The Chain Rule, essential for composite functions, states d/dx[f(g(x))] = f'(g(x)) · g'(x).
Derivatives of transcendental functions are equally important: d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[e^x] = e^x, and d/dx[ln x] = 1/x. Implicit differentiation allows us to find dy/dx when a relation is not explicitly solved for y. Higher-order derivatives (f'', f''') describe concavity and rates of change of rates.
Key terms
- Derivative
- The instantaneous rate of change of a function; the limit of the difference quotient as h approaches zero
- Chain Rule
- A rule for differentiating composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x)
- Implicit Differentiation
- A technique for finding dy/dx when y is defined implicitly as a function of x by differentiating both sides with respect to x
- Critical Point
- A point where f'(x) = 0 or f'(x) is undefined; candidates for local maxima and minima
- Second Derivative
- The derivative of the derivative, f''(x), which indicates the concavity of the original function
- Tangent Line
- The line that touches a curve at exactly one point with slope equal to the derivative at that point
Flashcards
- Power Rule
- d/dx[x^n] = nx^(n-1) for any real number n
- Product Rule
- d/dx[f·g] = f'g + fg'; differentiate first times second plus first times derivative of second
- Quotient Rule
- d/dx[f/g] = (f'g - fg') / g²; 'low d-high minus high d-low over low squared'
- Chain Rule
- d/dx[f(g(x))] = f'(g(x)) · g'(x); differentiate outer function keeping inner intact, then multiply by derivative of inner
- d/dx[sin x]
- cos x
- d/dx[ln x]
- 1/x, for x > 0
- Inflection Point
- A point where the concavity of f changes; occurs where f''(x) = 0 and changes sign
- Logarithmic Differentiation
- A technique using ln on both sides to differentiate complicated products, quotients, or variable exponents
Questions
Using the Chain Rule, find the derivative of f(x) = sin(3x²).
- 6x cos(3x²)
- cos(3x²)
- 6x sin(3x²)
- -6x cos(3x²)
Show answer
Answer: 6x cos(3x²)
By the Chain Rule: outer function is sin(u) where u = 3x². Derivative of sin(u) is cos(u), times derivative of inner 3x² which is 6x. Result: 6x cos(3x²).
The derivative of f(x) = e^x is f'(x) = xe^(x-1).
Show answer
Answer: False
The power rule applies to x^n, not e^x. The derivative of e^x is simply e^x — it is its own derivative.
Find dy/dx for y = x³ · ln(x) using the Product Rule.
- 3x² + x²
- 3x² ln(x) + x²
- x² ln(x) + 3x
- 3x² / ln(x)
Show answer
Answer: 3x² ln(x) + x²
Product Rule: d/dx[x³ · ln x] = 3x² · ln x + x³ · (1/x) = 3x² ln x + x².
Using the Power Rule, the derivative of f(x) = 7x⁵ is f'(x) = ___.
Show answer
Answer: 35x^4
Power Rule: d/dx[7x⁵] = 7 · 5x^(5-1) = 35x⁴.
If f'(c) = 0, then x = c is always a local maximum or minimum of f.
Show answer
Answer: False
f'(c) = 0 makes c a critical point, but it could also be an inflection point (e.g., f(x) = x³ at x = 0 has f'(0) = 0 but no local extremum).
Get a personalized course on Derivatives and Differentiation built around your child.