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]

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[pic]

So if Tan y = x, then

[pic]

[pic]

[pic]

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