Find $\frac{\mathrm d y}{\mathrm d x}$ if $y = \cos x$.

Analyze the question: You are asked to differentiate the cosine function with respect to x.

Concept: This is a standard derivative you must know for JAMB:

• d/dx (sin x) = cos x
• d/dx (cos x) = −sin x

Step-by-step solution:

We can confirm this from first principles:

1. dy/dx = lim (h→0) [cos(x+h) − cos x] / h
2. Expand cos(x+h) = cos x cos h − sin x sin h
3. So the expression becomes [cos x (cos h − 1) − sin x sin h] / h
4. Split it: cos x · (cos h − 1)/h − sin x · (sin h)/h
5. As h → 0, (cos h − 1)/h → 0 and (sin h)/h → 1
6. Therefore dy/dx = cos x(0) − sin x(1) = −sin x

Check the options:

• A. sin x: this is the derivative of −cos x, so it is wrong.
• B. −sin x: correct.
• C and D involve tan x, which does not come from differentiating cos x.

Answer: B. −sin x

Exam tip: Remember that the "co" functions have negative derivatives. The derivatives of cos x, cot x and cosec x all begin with a minus sign. Also, going around the cycle: sin → cos → −sin → −cos → sin. This is useful for higher derivatives too.