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MATH2B_Chapter4_Differentiation.pdf · Folio
MATH2B_Chapter4_Differentiation.pdf
p. 47 / 112
Chapter 4: Techniques of Differentiation

Differentiation is the process of finding the derivative of a function — the instantaneous rate of change with respect to its variable. Two fundamental approaches, explicit and implicit differentiation, differ in how the relationship between variables is expressed.

4.2 Implicit Differentiation

When y is defined implicitly through an equation in both x and y, differentiate both sides with respect to x, treating y as a function of x.

Key step: whenever you differentiate a y term, multiply by dy/dx (chain rule). Then isolate dy/dx.

Example 4.2

Find dy/dx for the equation x² + y² = 25.

Differentiating both sides: 2x + 2y(dy/dx) = 0  →  dy/dx = −x/y

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Ch. 4 · Calculus II
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What's the difference between implicit and explicit differentiation?
Explicit: y = f(x) — y is already isolated, apply rules directly.

Implicit: y and x are tangled together (like the circle on p. 47). You differentiate both sides, chain-rule any y term, then solve for dy/dx.
Give me 3 practice problems like example 4.2
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