Derivative of arcsin(x), inverse sine implicit differentiation + derivative of arcsin(1-x).

Опубликовано: 25 Август 2022
на канале: Zak's Lab

886

00:00 Compute the derivative of inverse sine: in order to make progress on the derivative of arcsin(x), we start by giving it a name, y. This allows us to invert the arcsine function and express the relationship as sin(y)=x. We differentiate both sides with respect to x, and this requires the chain rule! The derivative of sin(y) with respect to x means that the argument of the sine function is a non-trivial function of x. The chain rule says to differentiate first with respect to y, obtaining cos(y), then tack on the derivative of y with respect to x (dy/dx). We solve for dy/dx and obtain the derivative of inverse sine as 1/cos(y). The inverse sine implicit differentiation is complete, but we still need to write the final answer as a function of x!

01:21 We replace y with arcsin(x), but we still need to simplify cos(arcsin(x)). A trig function of an inverse trig function can always be simplified into algebraic form, and we do this by visualizing arcsin(x) (the angle whose sine is x) in a reference triangle. We label the angle y, and we label the opposite side and hypotenuse by using the fact that the sine of y is opposite over hypotenuse. Applying the pythagorean theorem, we find the adjacent side of the triangle is sqrt(1-x^2), which allows us to compute the cosine of y. After simplifying the cosine of inverse sine, we arrive at our formula for the derivative of arcsin(x) in terms of x: 1/sqrt(1-x^2).

03:10 Apply the derivative of arcsin formula to an example: in the example, we combine the arcsine derivative with the chain rule by using a more complicated argument of 1-x. We compute the derivative of arcsin(1-x). We start by differentiating with respect to 1-x and plugging into the arcsine derivative formula, then the chain rule requires us to multiply by the derivative OF 1-x, which is -1. Finally, we expand the square of (1-x) inside the square root and combine like terms to simplify the inside of the square root.