Question:
If tan $ y=\frac{sin\,x+cos\,x}{cos\,x-sin\,x} $ , then $ \frac{dy}{dx}= $
If tan $ y=\frac{sin\,x+cos\,x}{cos\,x-sin\,x} $ , then $ \frac{dy}{dx}= $
Updated On: Jun 23, 2024
- $ 1 $
- $ cosx $
- $ sinx $
- $ -1 $
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The Correct Option is A
Solution and Explanation
We have, $tan\,y = \frac{sin\,x + cos\,x}{cos\,x - sin\,x}$
$= \frac {1 + tan\,x}{1-tan\,x} = tan(\frac{\pi}{4} + x)$
$\Rightarrow y = tan^{-1} (tan(\frac{\pi}{4} + x))$
$ = \frac {\pi}{4} + x$
Differentiating w.r.t. $x$, we get
$\frac{dy}{dx} = 1$
$= \frac {1 + tan\,x}{1-tan\,x} = tan(\frac{\pi}{4} + x)$
$\Rightarrow y = tan^{-1} (tan(\frac{\pi}{4} + x))$
$ = \frac {\pi}{4} + x$
Differentiating w.r.t. $x$, we get
$\frac{dy}{dx} = 1$
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.