y=cos2x, then \(\frac{dy}{dx}\) is______
- tan2x
- cos2x
- cot2x
- -sin2x
The Correct Option is D
Solution and Explanation
To find the derivative \(\frac{dy}{dx}\) of the function \(y = \cos^2 x\), we'll use the chain rule of differentiation.
1. Given Function:
\[
y = \cos^2 x
\]
This can be rewritten as:
\[
y = (\cos x)^2
\]
2. Apply the Chain Rule:
The chain rule states that if \(y = [f(x)]^n\), then:
\[
\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)
\]
Here, \(f(x) = \cos x\) and \(n = 2\).
3. Differentiate Step-by-Step:
\[
\frac{dy}{dx} = 2(\cos x)^{2-1} \cdot \frac{d}{dx}(\cos x) \\
= 2\cos x \cdot (-\sin x) \\
= -2\sin x \cos x
\]
4. Simplify Using Trigonometric Identity:
We can express the result using the double-angle identity:
\[
\sin 2x = 2\sin x \cos x
\]
Thus:
\[
\frac{dy}{dx} = -\sin 2x
\]
5. Final Answer:
The derivative of \(y = \cos^2 x\) with respect to \(x\) is:
\[
\boxed{-2\sin x \cos x} \quad \text{or equivalently} \quad \boxed{-\sin 2x}
\]
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Concepts Used:
Differentiability
Differentiability of a function A function f(x) is said to be differentiable at a point of its domain if it has a finite derivative at that point. Thus f(x) is differentiable at x = a
\(\frac{d y}{d x}=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h}=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)
⇒ f'(a – 0) = f'(a + 0)
⇒ left-hand derivative = right-hand derivative.
Thus function f is said to be differentiable if left hand derivative & right hand derivative both exist finitely and are equal.
If f(x) is differentiable then its graph must be smooth i.e. there should be no break or corner.
Note:
(i) Every differentiable function is necessarily continuous but every continuous function is not necessarily differentiable i.e. Differentiability ⇒ continuity but continuity ⇏ differentiability
(ii) For any curve y = f(x), if at any point \(\frac{d y}{d x}\) = 0 or does not exist then, the point is called “critical point”.
3. Differentiability in an interval
(a) A function fx) is said to be differentiable in an open interval (a, b), if it is differentiable at every point of the interval.
(b) A function f(x) is differentiable in a closed interval [a, b] if it is
- Differentiable at every point of interval (a, b)
- Right derivative exists at x = a
- Left derivative exists at x = b.