Find \(\frac {dy}{dx}\): \(x^3+x^2y+xy^2+y^3=81\)
Solution and Explanation
The given relationship is x3 + x2y + xy2 + y3 = 81
Differentiating this relationship with respect to x, we obtain
\(\frac {d}{dx}\)(x3 + x2y + xy2 + y3) = \(\frac {d}{dx}\)(81)
⇒ \(\frac {d}{dx}\)(x3) + \(\frac {d}{dx}\)(x2y) + \(\frac {d}{dx}\)(xy2) + \(\frac {d}{dx}\)(y3)=0
⇒ 3x2 + [y.\(\frac {d}{dx}\)(x2) + x2.\(\frac {dy}{dx}\)] + [y2 \(\frac {d}{dx}\)(x) +x \(\frac {d}{dx}\)(y2)] + 3y2\(\frac {d}{dx}\) = 0
⇒ 3x2 + [y.2x + x2\(\frac {dy}{dx}\)] + [y2.1 + x.2y.\(\frac {dy}{dx}\)] + 3y2\(\frac {dy}{dx}\) = 0
⇒ (x2 + 2xy + 3y2)\(\frac {dy}{dx}\) + (3x2 + 2xy + y2) = 0
∴ \(\frac {dy}{dx}\) = \(-\frac {(3x^2+2xy+y^2)}{(x^2+2xy+3y^2)}\)
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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.