Derivatives
Grade 12 · Math · Study Guide
Understand the derivative as a rate of change and master the core differentiation rules.
The derivative f′(x) measures the instantaneous rate of change of f at x. Geometrically, it equals the slope of the tangent line to the graph at that point. Formally, f′(x) = lim(h→0) [f(x+h) - f(x)] / h.
Several shortcut rules make differentiation efficient. The power rule states d/dx[xⁿ] = nxⁿ⁻¹. The product rule handles products: d/dx[fg] = f′g + fg′. The quotient rule gives d/dx[f/g] = (f′g - fg′)/g².
The chain rule is used when differentiating composite functions: d/dx[f(g(x))] = f′(g(x)) · g′(x). Recognizing the 'outer' and 'inner' functions is the key skill.
| Rule | Formula |
|---|---|
| Power | d/dx[xⁿ] = nxⁿ⁻¹ |
| Constant | d/dx[c] = 0 |
| Product | d/dx[fg] = f′g + fg′ |
| Quotient | d/dx[f/g] = (f′g − fg′) / g² |
| Chain | d/dx[f(g(x))] = f′(g(x)) · g′(x) |
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| eˣ | eˣ |
| ln x | 1/x |
| tan x | sec² x |
Key terms
- Derivative
- The instantaneous rate of change of a function; slope of the tangent line.
- Differentiable
- A function is differentiable at a point if its derivative exists there.
- Chain Rule
- Differentiation rule for composite functions: multiply derivative of outer by derivative of inner.
- Implicit Differentiation
- Technique to differentiate equations not solved for y, treating y as a function of x.
Practice questions
What is d/dx[3x⁴ - 2x² + 5]?
- 12x³ - 4x
- 12x³ - 2x
- 12x⁴ - 4x
- 3x³ - 2x
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Apply the power rule term by term: 4·3x³ - 2·2x + 0 = 12x³ - 4x.
Find the derivative of f(x) = sin(x²).
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Chain rule: outer = sin(u), inner = x². f′ = cos(x²) · 2x = 2x cos(x²).
Using the product rule, what is d/dx[x² · eˣ]?
- 2x · eˣ
- x² · eˣ
- 2x · eˣ + x² · eˣ
- 2x + eˣ
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Product rule: (x²)′eˣ + x²(eˣ)′ = 2xeˣ + x²eˣ.
If f is continuous at x = a, then f must be differentiable at x = a.
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f(x) = |x| is continuous at 0 but not differentiable there (sharp corner).
Find d/dx[ln(3x)].
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ln(3x) = ln 3 + ln x. d/dx[ln 3] = 0, d/dx[ln x] = 1/x. Total: 1/x.
Fun facts
- Newton and Leibniz independently invented calculus in the 17th century, sparking one of history's biggest academic disputes over credit.
- The derivative of eˣ is itself — eˣ is the only function (up to a constant) with this unique property.
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