Answer:
[tex]\frac{d(cos^{-1}x )}{dx} = \frac{-1}{\sqrt{1-x^2} }[/tex]
Step-by-step explanation:
Given the differential (d/dx)(cos−1(x)), to find the equivalent formula we will differentiate the inverse function using chain rule as shown below;
let;
[tex]y = cos^{-1} x \\\\taking \ cos\ of\ both\ sides\\\\cosy = cos(cos^{-1} x)\\\\cosy = x\\\\x = cosy\\\\\frac{dx}{dy} = -siny\\[/tex]
[tex]\frac{dy}{dx} = \frac{-1}{sin y} \\\\from\ trigonometry\ identity,\ sin^{2} x+cos^{2}x = 1\\sinx = \sqrt{1-cos^{2} x}[/tex]
Therefore;
[tex]\frac{dy}{dx} = \frac{-1}{\sqrt{1-cos^{2}y } }[/tex]
Since x = cos y from the above substitute;
[tex]\frac{dy}{dx} = \frac{-1}{\sqrt{1-x^{2}} }[/tex]
Hence, [tex]\frac{d(cos^{-1}x )}{dx} = \frac{-1}{\sqrt{1-x^2} }[/tex] gives the required proof
