Find \(\frac {dy}{dx}:\) \(sin^2x+cos^2y=1\)
Solution and Explanation
The given relationship is sin2x + cos2y = 1
Differentiating this relationship with respect to x, we obtain
\(\frac {d}{dx}\)(sin2x + cos2y) = \(\frac {d}{dx}\)(1)
\(\implies\)\(\frac {d}{dx}\)(sin2x) + \(\frac {d}{dx}\)(cos2y) = 0
\(\implies\)2sin x . \(\frac {d}{dx}\)(sin x) + 2cos y . \(\frac {d}{dx}\)(cos y) = 0
\(\implies\)2sin x . cos x + 2cos y (-sin y) . \(\frac {dy}{dx}\) = 0
\(\implies\)sin2x - sin2y\(\frac {dy}{dx}\) = 0
∴\(\frac {dy}{dx}\) = \(\frac {sin\ 2x} {sin\ 2y}\)
Top Questions on Continuity and differentiability
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Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
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What is the Planning Process?
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Notes on Continuity and differentiability
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.