Find the derivative of θ/θ

Correct Answer is : sec2 θθθ

Correct Answer is : sec2 θθθ

Test Index

JAMB Mathematics 2011 Paper

Question : 43

Total: 46

\frac{\sin\theta}{\cos\theta}

Solution:

Concept:

The given expression is

\frac{\sin\theta}{\cos\theta}

, which equals

\tan\theta

Explanation:

Let

y = \frac{\sin\theta}{\cos\theta}

Apply the quotient rule:

y' = \frac{\cos\theta \cdot \frac{d}{d\theta}(\sin\theta) - \sin\theta \cdot \frac{d}{d\theta}(\cos\theta)}{\cos^2\theta}

We have

\frac{d}{d\theta}(\sin\theta) = \cos\theta

and

\frac{d}{d\theta}(\cos\theta) = -\sin\theta

Substitute these derivatives:

y' = \frac{\cos\theta \cdot \cos\theta - \sin\theta(-\sin\theta)}{\cos^2\theta}

Simplify the numerator:

y' = \frac{\cos^2\theta + \sin^2\theta}{\cos^2\theta}

Using the identity

\cos^2\theta + \sin^2\theta = 1

y' = \frac{1}{\cos^2\theta}

Since

\frac{1}{\cos\theta} = \sec\theta

, we get:

y' = \sec^2\theta

Answer:

\sec^2\theta

Option A.

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