Derivatives of Inverse Functions – Notes
Derivatives of Inverse Functions – Notes (1)
1. [pic]
(a) If [pic].
Recall [pic] and [pic], so [pic].
From the Derivative of an Inverse Theorem, we have: [pic]
So [pic]. Recall: [pic]
(b) Use implicit differentiation to find the derivative of the inverse of y = ex.
Here we’ll use an old trick exchanging the x & y to find the inverse.
So we begin with: [pic]. Now take the derivative w.r.t. ‘x’.
[pic]
Substituting x for ey (or keeping in mind that y = ln x)…
[pic]
2. [pic]
(a) If [pic]
Recall [pic] and [pic]
So [pic]
(b) Use implicit differentiation to get the derivative of the inverse of [pic]
Exchange the ‘x’ and the ‘y’ to get: x = Sin y. Take the derivative w.r.t. x:
[pic]. Now we pull out that ‘Rt Triangle Trick Trick’!
If sin y = x, what is cos y = ? (See figure below) cos y = [pic]= [pic]
[pic] dy/dx = 1/cos y or [pic]
Derivatives of Inverse Functions – Notes (2)
3. Given [pic] with [pic],
(a) Find [pic]
[pic] and [pic]
[pic] and then we’ll use: [pic]
to get: [pic]
(b) Use implicit differentiation to find [pic]
To find the inverse function for y = Tan x, we exchange the ‘x’ and the ‘y’ to get:
[pic] and then we take the derivative w.r.t. ‘x’ to get:
[pic] and then we use the rt [pic] trig trick:
[pic]
4. Derivatives of the other Inverse Trig Functions
[pic] and [pic]
both of which can be proven by taking the derivative w.r.t. ‘x’ of the trig identities:
[pic] and [pic]
We don’t worry too much about Inverse Secant and Inverse Cosecant since we can
Always use the following identities involving Inverse Cosine and Inverse Sine:
[pic] and [pic]
By the way… this doesn’t quite work for Inverse Cotangent, since:
[pic]
ex/ [pic] but… ex/ [pic]
-----------------------
[pic]
So if Tan y = x, then
[pic]
[pic]
[pic]
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