The derivative of $\sin x^{\circ} \, \, \cos x$ with respect to $x$ is
- $\frac{\pi}{180} (\cos \, x^{\circ} \, \cos \, x - \sin \, x^{\circ} \sin \, x)$
- $\frac{\pi}{180} \cos \, x^{\circ} \, \cos \, x - \sin \, x^{\circ} \sin \, x$
- $\frac{180}{\pi} (\cos \, x^{\circ} \, \cos \, x - \sin \, x^{\circ} \sin \, x)$
- $- \frac{\pi}{180} \cos \, x^{\circ} \, \sin \, x $
The Correct Option is B
Solution and Explanation
First we change $x^{\circ}$ into radian
$i.e.,$ 1 degree $ = \frac{\pi}{180} $ radian
$\Rightarrow \, \, \, x^{\circ} = \frac{\pi}{180} x$ redian
$ \therefore \, \, \, \, \sin x^{\circ} . \cos x = \sin \left(\frac{\pi}{180} x \right) \cos x$
Now differentiating w.r.t. $'x'$, we have
$\Rightarrow \left[\cos\left(\frac{\pi}{180} x\right).\left(\frac{\pi}{180}\right) \cos x\right] +\sin\left(\frac{\pi}{180}x\right)\left(-\sin x\right)$
$ \Rightarrow \frac{\pi}{180} \cos x^{\circ} \cos x -\sin x^{\circ} \sin x$
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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.