# Tagged Questions

49 views

### How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
31 views

### How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
69 views

### I need help finding the derivative of the inverse function.

So $$f(x)=\frac{x+1}{2x-1}$$ and $$g(x)$$ is an inverse of $$f(x)$$ I have the points on $f(x)$ of (2,1). So I know that $f(2)=1$, $g(1)=2$ and $g'(1)=\frac{1}{f'[g(1)]}$ so $g'(1)=\frac{1}{f'(2)}$ ...
38 views

### Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
39 views

### Element-wise derivative of the inverse of a matrix

I would appreciate if you could help me to obtain the element-wise derivative of $Z = (-A-BX)^{(-1)}$ where all of elements of $A$, $B$ and $X$ are positive. I conjecture that if I increase any of ...
19 views

### Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
40 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. ...
46 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? ...
39 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 ...
68 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 ...
128 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 ...
31 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 ...
88 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) ...
47 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)
91 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 ...
113 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 ...
26 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. ...
101 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 ...
26 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 ...
88 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 ...
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 ...
116 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 ...
92 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 ...
108 views

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 $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ... 1answer 110 views ### A formula for n-derivative of the inverse of a function? Let$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} ... 2answers 435 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 ... 1answer 92 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 ... 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 ... 2answers 115 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. 2answers 340 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 ... 3answers 2k 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$\frac{d}{dx}f^{-1}(3)$... 1answer 250 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 ... 1answer 87 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$f:U\to\mathbb{R}^{n}$... 2answers 846 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]' = ...
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 ...