How to Find Derivatives of Inverse Functions from the Table?
Let us discuss this concept with the help of an example. So let us assume g and f be the inverse function and the following table lists a few values of f, g and f’.
| x | f(x) | g(x) | f'(x) |
|---|---|---|---|
| 2 | 4 | 8 | [Tex]\frac{-1}{6}[/Tex] |
| 8 | 3 | 2 | [Tex]\frac{1}{2}[/Tex] |
We have to find g'(2). As it said from the question, that f and g be inverse functions. This means if we have two sets, now let us assume that the first set is the domain of f. So in this set, if we start from some x value then f is going to map that x to another value which is known as f(x)(this is the use of function f). Now as we know that g is the inverse of f, so this g gets us back to the first set(this is the use of function g).
Hence we get
g(f(x)) = x …(i)
f(g(x)) = x …(ii)
Where, both are valid.
From eq(ii), we have
f(g(x)) = x
Now differentiate both side w.r.t. x. we get
[Tex]\frac{d(f(g(x)))}{dx} = \frac{d(x)}{dx}[/Tex]
Now on the LHS we apply chain rule, now we get
f'(g(x))g'(x) = 1
g'(x) = [Tex]\mathbf{\frac{1}{f'(g(x))}} [/Tex]
Now we are going to find the value of g'(2)
g'(2) = [Tex]\frac{1}{f'(g(2))}[/Tex]
From the table we get the value of g(2)
g'(2) = [Tex]\frac{1}{f'(8)}[/Tex]
From the table we get the value of f'(8)
g'(2) = [Tex]\frac{1}{\frac{1}{2}}[/Tex]
g'(2) = 2
Hence, the value of g'(2) = 2.
Derivatives of Inverse Functions
In mathematics, a function(e.g. f), is said to be an inverse of another(e.g. g), if given the output of g returns the input value given to f. Additionally, this must hold true for every element in the domain co-domain(range) of g. E.g. assuming x and y are constants if g(x) = y and f(y) = x then the function f is said to be an inverse of the function g. Or in other words, if a function f : A ⇢ B is one – one and onto function or bijective function, then a function defined by g : B ⇢ A is known as inverse of function f. The inverse function is also known as the anti function. The inverse of function is denoted by f-1.
f(g(x)) = g(f(x)) = x
Here, f and g are inverse functions.
Table of Content
- Overview of Derivatives of Inverse Functions
- Procedure of finding inverse of f
- Derivatives of Inverse Functions
- How to find derivatives of inverse functions from the table?
- Derivatives of Inverse Trigonometric Functions
- How to find the derivatives of inverse trigonometric functions?
- 1. Derivative of f given by f(x) = sin–1 x.
- 2. Derivative of f given by f(x) = cos–1 x.
- 3. Derivative of f given by f(x) = tan–1 x.
- 4. Derivative of f given by f(x) = cot–1 x.
- 5. Derivative of f given by f(x) = sec–1 x.
- 6. Derivative of f given by f(x) = cosec–1 x.
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