Derivatives of Inverse Trigs via Implicit Differentiation
We can use implicit differentiation to find derivatives of
inverse
functions. Recall that the equation y=f−1(x) means the
same things as
x=f(y).
Taking derivatives of both sides gives
ddxx=ddxf(y) and using the chain
rule we get 1=f′(y)dydx.
Dividing both sides by f′(y) (and swapping sides) gives
dydx=1f′(y). Once we rewrite f′(y) in
terms of
x, we have the derivative of f−1(x).
In the following video, we use this trick to differentiate
the inverse trig functions sin−1, cos−1 and
tan−1.
The end results are
ddxsin−1(x)=1√1−x2,ddxcos−1(x)=−1√1−x2,ddxtan−1(x)=11+x2.
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You should learn these derivatives; however, they will be most
helpful in the second semester of calculus.
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