Derivative of arcsin(x) For a ﬁnal example, we quickly ﬁnd the derivative of y = sin−1 x = arcsin x. As usual, we simplify the equation by taking the sine of both sides: y = sin−1 x sin y = x We next take the derivative of both sides of the equation and solve for y � = sin y = x � = 1 y� = 1 cos y (cos y) · y dy dx . We want to rewrite this in terms of x = sin y. Luckily there is a simple trig. identity relating cos y to sin y. We can solve it for cos y and “plug in”. cos2 y + sin2 y = cos2 y = cos y 1 1 − sin2 y � = 1 − sin2 y (cos y > 0 on the range of y = sin−1 x) Plugging this in to our equation for y � = y� = d dx sin−1 x we get: 1 1 1 =� =√ . 2 cos y 1 − x2 1 − sin y Notice that we made a choice between a positive and negative square root when solving for cos y. We chose the positive square root because we usually deﬁne sin−1 x to have outputs between −π/2 and π/2, and the cosine function is always positive on this interval. When dealing with inverse functions we are often faced with choices like this; when in doubt draw a graph and be sure your choices make sense in the context of your problem. 1