Question

If \[y=\sin (2{{\sin }^{-1}}x)\], then \[\dfrac{dy}{dx}=\]
A.\[\dfrac{(2-4{{x}^{2}})}{\sqrt{(1-{{x}^{2}})}}\]
B.\[\dfrac{(2+4{{x}^{2}})}{\sqrt{(1-{{x}^{2}})}}\]
C.\[\dfrac{(2-4{{x}^{2}})}{\sqrt{(1+{{x}^{2}})}}\]
D.None of these

Answer 1

Hint: To find the derivative we can use the slope formula, that is, \[slope=\dfrac{dy}{dx}\].
\[dy\]is the changes in \[Y\]and \[dx\] is the changes in \[X\]

To find the derivative of two variables in multiplication, this formula is used
\[\dfrac{d}{dx}(uv)=u\dfrac{d}{dx}(v)+v\dfrac{d}{dx}(u)\]
Mechanically, \[\dfrac{d}{dx}(f(x))\] measures the rate of change of \[f(x)\] with respect to \[x\] .
Differentiation of any constant is zero. Differentiation of constant and a function is equal to constant times the differentiation of the function. Geometrically, graph of a constant function is a straight line parallel to the \[x-axis\]. Consequently slope of the tangent is zero.

Complete step-by-step answer:
Given that \[y=\sin (2{{\sin }^{-1}}x)\]
Let us assume that \[x=\sin \theta \]
Differentiating both sides with respect to \[\theta \] we get
\[\dfrac{dx}{d\theta }=\cos \theta \]
Substituting these values in the equation we get
\[y=\sin (2{{\sin }^{-1}}\sin \theta )\]
Solving the above equation we get
\[y=\sin 2\theta \]
Differentiating both sides with respect to \[\theta \] we get
\[\dfrac{dy}{d\theta }=2\cos 2\theta \]
We know that \[\dfrac{dy}{dx}=\dfrac{dy}{d\theta }\times \dfrac{d\theta }{dx}\]
Substituting the values in the equation we get
\[\dfrac{dy}{dx}=\dfrac{2\cos 2\theta }{\cos \theta }\]
Further solving by applying trigonometric identities we get
\[\dfrac{dy}{dx}=\dfrac{2(1-2{{\sin }^{2}}\theta )}{\sqrt{(1-{{\sin }^{2}}\theta )}}\]
Substituting \[x=\sin \theta \] in the above equation we get
\[\dfrac{dy}{dx}=\dfrac{2(1-2{{x}^{2}})}{\sqrt{1-{{x}^{2}}}}\]
Further solving we get
\[\dfrac{dy}{dx}=\dfrac{(2-4{{x}^{2}})}{\sqrt{(1-{{x}^{2}})}}\]
Therefore, option \[A\] is the correct answer.
So, the correct answer is “Option A”.

Note: \[\dfrac{d}{dx}(f(x))=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f(x+h)-f(x)}{h}\] is the formula for finding the derivative from the first principles. The slope is the rate of change of \[y\] with respect to \[x\] that means if \[x\] is increased by an additional unit the change in \[y\] is given by \[\dfrac{dy}{dx}\] . Let us understand with an example, the rate of change of displacement of an object is defined as the velocity \[Km/hr~\] that means when time is increased by one hour the displacement changes by \[Km\]. For solving derivative problems different techniques of differentiation must be known.