Differentiation Techniques

This lesson covers the chain rule, product rule, quotient rule, and implicit differentiation — extending your differentiation toolkit for more complex functions.

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The Chain Rule

If y = f(g(x)), then:

dy/dx = f’(g(x)) × g’(x)

Or, if y = f(u) and u = g(x):

dy/dx = (dy/du) × (du/dx)

Worked Example 1

Differentiate y = (3x + 2)⁵.

Let u = 3x + 2, so y = u⁵.

dy/du = 5u⁴, du/dx = 3

dy/dx = 5(3x + 2)⁴ × 3

= 15(3x + 2)⁴

Worked Example 2

Differentiate y = sin(x²).

dy/dx = cos(x²) × 2x

= 2x cos(x²)

Worked Example 3

Differentiate y = e^(3x²+1).

dy/dx = e^(3x²+1) × 6x

= 6x e^(3x²+1)

Worked Example 4

Differentiate y = ln(2x + 5).

dy/dx = 1/(2x + 5) × 2

= 2/(2x + 5)

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The Product Rule

If y = u × v, where u and v are both functions of x:

dy/dx = u (dv/dx) + v (du/dx)

Worked Example 5

Differentiate y = x² sin x.