0
votes
0answers
17 views

Inverse functions determination by integral

From "Inverse functions and differentiation": Integrating this relationship gives $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. $$ This is only useful if the integral exists. ...
0
votes
1answer
27 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
0
votes
2answers
31 views

Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
0
votes
2answers
67 views

How do I find $(f^{-1})'(a)$? [closed]

if $$f(x) = 3x^3 + 3x^2 + 6x + 9 $$ $$a = 9$$ and also $$f(x) = 2x^3 + 3\sin x + 3\cos x$$ $$a = 3$$ I know I have to find the inverse but I think I’m getting overly complicated answers and my ...
1
vote
1answer
50 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
1
vote
2answers
28 views

Could someone please help me find the derivative of the inverse of $f$ at $0$?

The problem is: for $\displaystyle f(x)= \int_0^{\ln x} \frac{1}{\sqrt{4+\mathrm{e}^{t}}} \, \mathrm{d}t$, $x > 0$, find $(f^{-1})'(0)$. I know that I should use the fundamental theorem of ...
1
vote
1answer
54 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
0
votes
1answer
36 views

Could someone please help me solve this calculus problem?

For f(x)= integral 1/sqrt(4+e^t) dt from 0 to lnx, with x>0, find (f^-1)'(0) (that is, the derivative of the inverse of f at 0)
2
votes
5answers
71 views

Derivative of $ h(t)= \sin (\cos^{-1}t$)?

Can someone please explain the steps/rules I need to preform to find the derivative of $h(t)= \sin (\cos^{-1}t)$? I tried to used the product rule, and realized it was obviously a failure. Using ...
0
votes
3answers
72 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
0
votes
1answer
21 views

Chain rule with inverse function

In a proof, my professor shows: $ s = g^{-1}(u) $ $ ds = \frac{dg^{-1}(u)}{du} du $ , by the chain rule If I were to apply the chain rule to calculate ds, I would not get the du in the denominator. ...
1
vote
1answer
89 views

Inverse function of $y=2x+\sin x$

I was doing a long exercise when come to this point: calculate the inverse function of $y=2x+\sin x (x \in\mathbb R) $ and its derivative. I know that the derivative of an inverse function is ...
0
votes
1answer
23 views

Parametric Curves Existence of Tangent

If $\frac{dy}{dt}$ and $\frac{dx}{dt}$ exist, then does $\frac{dy}{dx}$ always exist when $\frac{dx}{dt} \not=0$? Indeed, this is a very simple question. Sorry but I'm just a beginner for ...
1
vote
1answer
84 views

Finding the inverse function

The question is to find the inverse function of $$f(x)=x-(2\sqrt{x})+1$$ I first found that the domain of definition is $\,x\ge 0$ Then studied the variation of the function and it is decreasing ...
1
vote
1answer
70 views

derivative of product of 2 inverse matrices

I was trying to differentiate the equation below: $$ \frac{\partial a^T X^{-T}X^{-1}a} {\partial X} $$ where X is invertible but not symmetric and $X^{-T}$ means transpose of inverse of X. In the ...
0
votes
1answer
77 views

Prove (local) converse to the implicit function theorem

The implicit function theorem tells us that: Given a level set $M^k = F^{-1}(F(p_0))$ of a smooth function $$F: \mathbb{R}^n \to \mathbb{R}^{n-k},$$ where $\operatorname{rk}{(Df)(p)} = n-k$ for ...
2
votes
3answers
84 views

Using both Leibniz' notation and prime-notation for a derivative

I am presented with the following task: "Assume that the function $f(x)$ has the derivative $f'(x) = \frac{1}{x}$ and that $f$ is one-to-one. If $y = f^{-1}(x)$, show that $\frac{dy}{dx} = 1$. The ...
3
votes
2answers
91 views

Formula for Nth Derivative of Matrix Inverse

I was looking for an equation for the nth derivative of a matrix inverse, ie $\frac{d^n \bf{A}^{-1}}{dx^n}$ I know that the first derivative $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ...
3
votes
1answer
98 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
1
vote
2answers
289 views

Derivative of Standard Normal Inverse

How can I calculate the derivative of the standard normal inverse. I think the derivative of $\Phi^{-1}(x)$ is $$\frac{1}{\phi(\Phi^{-1}(x))}.$$ I would like to know how to find the derivative of ...
2
votes
1answer
83 views

Derivative of the inverse of $y=(a+bx)e^{cx}$

I need to solve for the 1st derivative of the inverse of $y=(a+bx)e^{cx}$ but my calculus is a bit rusty. I know that to get the inverse function, I would have to use the Lambert W method but I think ...
2
votes
2answers
40 views

How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$

Question: Let $E=\{(x,y): 0<y<x \}$ set $f(x,y)=(x+y, xy)$ for $(x,y)\in E$ a) How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$ And how to find formula for ...
5
votes
2answers
109 views

Inverse and derivative of a function [duplicate]

Find an example of an inverse function f(x) such that its derivative is the same as its inverse. I tried many different functions but non of them worked.
0
votes
2answers
270 views

Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$

Given the matrix equation: $$ x^TA^TA = b^TA $$ I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric. When I solve it ...
3
votes
3answers
1k views

Evaluate the derivative of an inverse function by using a table of values?

Given the function and derivative values in the table below, evaluate $\frac{d}{dx}f^{-1}(3)$ ...
2
votes
1answer
233 views

Identify the equation of the normal line?

Identify the equation of the normal line to the curve $y=g(p)=2.5+3.5(4^p)$ where it crosses the $y$-axis. So I am guessing the normal line would be the inverse of the derivative function, since it ...
1
vote
1answer
83 views

A differentiable injective function with Lipschitzian Inverse

I'm having difficulty with the following question which was given to me following studying the inverse mapping theorem. Let $U\subseteq\mathbb{R}^{n}$ be an open set and let $f:U\to\mathbb{R}^{n}$ ...
2
votes
2answers
686 views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = ...
7
votes
0answers
727 views

Functions whose derivative is the inverse of that function

Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?) I was drawing some polynomials and their ...
1
vote
1answer
156 views

How to find the frechet derivative at $A\rightarrow A^{-1}$ mapping?

I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) . The 1.1.3 (b) is : Let F: $U\rightarrow ...
3
votes
1answer
135 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
0
votes
2answers
135 views

Let $m$ be the inverse function of $h(x) = 3x + \cos(2x)$. Find $m'(\frac{3\pi}{4})$

I wasn't sure how to solve for $x$ and create the inverse function given the $\cos(2x)$ term. Whenever I tried to take $\arccos$ there was an $x$ on the other side of the equality which meant I was ...
2
votes
1answer
617 views

Differentiate an Inverse Function, Two Methods?

I would like to take the derivative of this inverse function at $\pi$: $f(x) = 2x + \cos{x}$, given that ${f}^{-1}(\pi) = \frac{\pi}{2}$. I know that there are two methods of doing it. Let me ...
0
votes
1answer
435 views

Derivative of a complicated inverse function

$\Phi(\cdot,0,1)$ and $\phi(\cdot,0,1)$ are cdf and pdf of standard normal distribution. $$y=F_\text{mix}(x,\mu,\sigma)=\sum\limits_{i=1}^{K}\lambda_i\Phi\left(\frac{x-\mu_i}{\sigma_i},0,1\right).$$ ...