Differentiation Basics

Apply the power rule fast and accurately for polynomial functions.

Difficulty: Easy → Medium Keywords: differentiation, power rule, derivative, SPM Add Maths
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Overview

Differentiation tells you how fast a function changes. In SPM Additional Mathematics, most early differentiation questions focus on polynomials, which means the power rule is your best friend. The power rule says that the derivative of ax^n is a·n·x^{n-1}. Apply it to each term independently, drop constants, and simplify. Accuracy matters because a small sign error can derail the entire question. This topic builds speed and consistency with step-by-step practice: read the function, apply the rule, and tidy the result. Once you are confident, you can handle tangents, optimization, and rates of change with far less stress. A smart check is to compare the degree: the derivative of a polynomial should be one degree lower. If it is not, recheck your work. Review the worked examples, then try a short practice set to make the power rule automatic.

Worked Examples

Differentiate: $$ f(x)=3x^2-4x+7 $$
  1. Apply the power rule term-by-term.
  2. Derivative: $$ f'(x)=6x-4 $$
$$ 6x-4 $$
Differentiate: $$ y=5x^3+2x $$
  1. Derivative of 5x^3 is 15x^2.
  2. Derivative of 2x is 2.
$$ 15x^2+2 $$
Differentiate: $$ y=7x^5-3x^3+4x $$
  1. Apply the power rule to each term.
  2. Result: $$ 35x^4-9x^2+4 $$
$$ 35x^4-9x^2+4 $$

FAQ

Do constants always disappear?

Yes. The derivative of any constant is 0.

What happens to the power?

You multiply by the original power and reduce the power by one.

Can I differentiate term-by-term?

Absolutely. That is the standard method for polynomials.