- #1
Kyoma
- 97
- 0
d/dx (sin-1x) = [itex]\frac{1}{\sqrt{1-x2}}[/itex]
But using chain rule, I got:
dy/dx = -sin-2xcosx
Why?
But using chain rule, I got:
dy/dx = -sin-2xcosx
Why?
Kyoma said:d/dx (sin-1x) = [itex]\frac{1}{\sqrt{1-x2}}[/itex]
But using chain rule, I got:
dy/dx = -sin-2xcosx
Why?
The notation d/dx represents the derivative of a function with respect to the independent variable x. In the case of sin-1x, it represents the derivative of the inverse sine function with respect to x.
To derive this equation, we can use the chain rule and the derivative of the inverse trigonometric function. First, we rewrite sin-1x as arcsin(x). Then, using the chain rule, we get dy/dx = d/dx(sin(arcsin(x))). By applying the derivative of the inverse trigonometric function, we get dy/dx = cos(arcsin(x)). Finally, using the Pythagorean identity sin2x + cos2x = 1, we can rewrite cos(arcsin(x)) as √(1-x2), giving us dy/dx = √(1-x2). Since sin-1x represents the inverse of sinx, we can replace sinx with x to get dy/dx = √(1-x2) = √(1-x2) * 1 = √(1-x2) * sin-2x, which simplifies to dy/dx = -sin-2xcosx.
The domain of sin-1x is [-1, 1], which means that the input value x can range from -1 to 1. The range of sin-1x is [-π/2, π/2], which means that the output value of arcsin(x) can range from -π/2 to π/2. This is because the inverse sine function only returns values within this range.
The graph of sin-1x is the inverse of the graph of sinx. This means that the graph of sin-1x is a reflection of the graph of sinx over the line y=x. The graph of sin-1x is also a curve that increases from -π/2 to π/2, similar to the graph of sinx.
The derivative of sin-1x can be used to solve problems in physics, engineering, and other fields that involve the inverse sine function. For example, it can be used to find the slope of a curve, which is critical in determining the velocity and acceleration of an object in motion. It can also be used in calculating the rate of change of various physical quantities, such as temperature or pressure, in different scenarios.