3.8 Derivatives of Inverse Trig Fcns.notebook
AP Calculus
3.8 Derivatives of Inverse Trigonometric Equations, p. 165
Derivatives of Inverse Functions
This means that the we can find the derivative of an inverse function as long
as the derivative of the original function does not equal zero on a given
interval.
We know that an inverse function will "undo" a function, i.e.:
Apply the chain rule to this equation, and solve for the derivative of the inverse:
If you truly understand what an inverse is, you can
always work this formula out from first principles.
Mr. Schindelka
1
3.8 Derivatives of Inverse Trig Fcns.notebook
AP Calculus
Things you need to remember about trigonometric functions:
The unit circle (radius = 1) has
coordinates that are related by the
Pythagorean theorem. Show how
you can express every (x,y)
coordinate in terms of sine and
cosine functions.
Graph y=sinx
Graph y=cosx
Graph the function y = tan θ on [0, 2 ]
Recall that the tangent function is the
slope of the radius of the unit circle.
Mr. Schindelka
2
3.8 Derivatives of Inverse Trig Fcns.notebook
AP Calculus
Working out the Derivatives of Inverse Trigonometric Functions
The trig functions are not one-to-one, so they do not have inverse functions. We can overcome this by restricting their domains.
To find the derivative of the inverse of y=sinx we need an interval where the
derivative of sinx is never zero. (Note: the inverse of sinx is known as arcsinx.)
y=sinx
Now find the derivative of y=sin-1x on this interval.
y=sin-1x
(rearrange and use implicit differentiation)
Now remember what y represents, and use the unit
circle to find an expression for cosy in terms of x.
You can find the derivatives of all of the inverse trigonometric functions in a similar fashion.
Mr. Schindelka
3
3.8 Derivatives of Inverse Trig Fcns.notebook
AP Calculus
which means
ex. If x + y = tan-1(x2 + 3y), find the derivative with respect to x.
3.8 Ass't: p. 170 # 1, 5-8, 12, 14, 19, 25, 35-40
Mr. Schindelka
4