3. Derivatives of the Inverse Trigonometric Functions
by M. Bourne
Recall from when we first met inverse trigonometric functions:
"sin-1x" means "find the angle whose sine equals x".
If x = sin-10.2588 then by using the calculator, x = 15°. We have found the angle whose sine is 0.2588.
We also write: arcsin x to mean the same thing as sin-1x.
It is better to use arcsin x because normally in mathematics, a number raised to the power `-1` means the reciprocal. Example: `3^-1=1/3`.
Most calculators use the (confusing) notation: `sin^-1 x`. This section mostly uses the sin-1x notation (since it was originally written to be consistent with calculator notation), however you are encouraged to use the superior notation, `arcsin\ x`.
You may also wish to go back to background information on inverse trigonmetric equations.
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Derivatives of Inverse Trigonometric Functions
The following are the formulas for the derivatives of the inverse trigonometric functions:
Find the derivative of y = cos-15x.
Find the derivative of y = sin-1(1 − x2).
Find `(dy)/(dx)` if x + y = tan-1(x2 + 3y).
1. Find the derivative of y = 3 cos-1(x2 + 0.5).
2. Find the derivative of y = 4 tan-13x4.
3. Find the derivative of y = (x2 + 1) sin-14x.
4. Find the derivative of sin-1(x + y) + y = x2.