\(Find \ \frac {dy}{dx}:\)
\(y=sec^{-1}(\frac {1}{2x^2-1}),\ 0<x< \frac {1}{\sqrt2}\)
\(Find \ \frac {dy}{dx}:\)
\(y=sec^{-1}(\frac {1}{2x^2-1}),\ 0<x< \frac {1}{\sqrt2}\)
Solution and Explanation
The given relationship is y = sec-1\((\frac {1}{2x^2-1})\)
y = sec-1\((\frac {1}{2x^2-1})\)
\(⇒\)sec y = \(\frac {1}{2x^2-1}\)
\(⇒\)cos y = 2x2-1
\(⇒\)2x2 = 1+cos y
\(⇒\)2x2 = 2cos2\(\frac y2\)
\(⇒\)x = cos \(\frac y2\)
Differentiating this relationship with respect to x, we obtain
\(\frac {d}{dx}\)(x) = \(\frac {d}{dx}\)(cos\(\frac y2\))
\(⇒\)1 = -sin\(\frac y2\) . \(\frac {d}{dx}\)(\(\frac y2\))
\(⇒\)-\(\frac {1}{sin\frac y2}\) = \(\frac 12\)\(\frac {dy}{dx}\)
\(⇒\)\(\frac {dy}{dx}\)x = -\(\frac {2}{sin\frac y2}\)
\(⇒\)\(\frac {dy}{dx}\)x= -\(\frac {2}{\sqrt {1-cos^2 \frac y2}}\)
\(⇒\)\(\frac {dy}{dx}\)= \(\frac {-2}{\sqrt {1-x^2}}\)
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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.