Examples on Derivative of sin inverse x
Example 1: Find derivative of function represented as f(x) = sin-1(2x).
Solution:
We know that, (sin-1x)β = 1/β(1-x2)
By chain rule, we get,
β (sin-12x)β = 1/β(1-(2x)2) Γ d/dx(2x)
β (sin-12x)β = 2/β(1-(2x)2)
β (sin-12x)β = 2/β(1-4x2)
Thus, for f(x) = sin-1(2x), we get, f'(x) = 2/β(1-4x2)
Example 2: Find derivative of function, f(x) = sin-1 (x2).
Solution:
We have, f(x) = sin-1 (x2)
We know that (sin-1x)β = 1/β(1-x2)
Thus, by applying chain rule, we get,
β f'(x) = 1/β(1-x4) Γ d/dx(x2)
β f'(x) = [1/β(1-x4)]Γ2x
β f'(x) = 2x/β(1-x4)
Example 3: If p(x) = x2sin-1x, find p'(x).
Solution:
For function p(x), we see that two functions are given in product. Thus, we use product rule of differentiation to obtain p'(x).
β p'(x) = x2 Γ d/dx(sin-1x) + d/dx(x2) Γ sin-1x
β p'(x) = x2/β(1-x2) + 2xsin-1x
Thus, for p(x) = x2sin-1x, we have, p'(x) = x2/β(1-x2) + 2xsin-1x
Example 4: Determine slope of tangent drawn to curve represented by y = sin-1x at x = 1/β2.
Solution:
We know that slope to tangent at any curve is given dy/dx.
Thus, for y = sin-1x, we have,
dy/dx = 1/β(1-x2)
Now, slope of tangent to curve at x = 1/β2 would be,
(dy/dx) x = 1/β2 = 1/β (1-(1/β2)2)
(dy/dx) x = 1/β2 = β2
Thus, slope of tangent to curve y = sin-1x at x=1/β2 would be β2.
Example 5: Find the derivative of function given by f(x) = sin-1 (cos x).
Solution:
We know, (sin-1x)β = 1/β(1-x2) and (cos x)β = -sin x
Therefore, by applying chain rule of differentiation for f(x) = sin-1 (cos x), we get,
f'(x) = 1/β(1-cos2x)Γd/dx(cos x)
f'(x) = -sin x/sin x = -1
Thus, f'(x) comes out to be -1.
Alternatively,
f(x) = sin-1(cos x) = sin-1(sin (Ο/2 β x))
f(x) = Ο/2 β x
Thus, f'(x) = -1
Derivative of Sin Inverse x
Derivative of sin inverse x is 1/β(1-x2). The derivative of any function gives the rate of change of the functional value with respect to the input variable. Sin inverse x is one of the inverse trigonometric functions. It is also represented as sin-1x. There are inverse trigonometric functions corresponding to each trigonometric function. The derivative of a function also helps in finding the slope of the tangent to the curve represented by the function at any point.
In this article, we will learn about the derivative of sin inverse x, methods to find it including the first principle of differentiation and implicit differentiation, solved examples, and practice problems.
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