Differentiation techniques
Objective. Use product, quotient, chain and implicit differentiation.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Differentiation techniques extend the basic power rule to combinations of functions: the product rule (fg)'=f'g+fg', the quotient rule (f/g)'=(f'g−fg')/g², and the chain rule (f(g(x)))'=f'(g(x))·g'(x) for composite functions. Implicit differentiation differentiates both sides of an equation relating x and y with respect to x, treating y as a function of x and applying the chain rule to y-terms, then solving for dy/dx. For x²+y²=25, differentiating gives 2x+2y(dy/dx)=0, so dy/dx=−x/y. Choosing the correct rule — product, quotient or chain — depends on whether the expression is a product, ratio, or nested composition.
Worked example
Differentiate y=x²sin(x) with respect to x using the appropriate differentiation technique.
- 1Recognise this as a product f(x)g(x) with f=x², g=sin(x), requiring the product rule.
- 2Compute f'=2x and g'=cos(x).
- 3Apply the product rule: dy/dx = f'g + fg' = 2x·sin(x) + x²·cos(x).
- 4This is fully simplified since no like terms combine further.
Answer: dy/dx = 2x sin(x) + x² cos(x)
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Practice
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Start practisingCommon mistake. Students differentiate a product like x²sin(x) by multiplying the separate derivatives 2x·cos(x), forgetting the product rule requires f'g+fg'; the correct derivative is 2x sin(x)+x²cos(x).
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