Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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3
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1answer
80 views

Prove that $\frac{d^n}{dx^n} (\sin^4 x + \cos^4 x) = 4^{n-1}\cos (4x + \frac{n\pi}{2})$

Question Prove that $\frac{d^n}{dx^n} (\sin^4 x + \cos^4 x) = 4^{n-1}\cos (4x + \frac{n\pi}{2})$ My attempt First calculate $\frac{d}{dx} (\sin^4 x + \cos^4 x)$, that is, $$\frac{d}{dx} ...
1
vote
1answer
18 views

Help with simplifying implicit differentiation

Given the equation $\frac{y}{x+7y} = x^6 + 7$, find $\frac{dy}{dx}$. Ok, so I started to solve for $\frac{dy}{dx}$ and got to here: $\frac{\frac{dy}{dx}(x+7y)-(1+7\frac{dy}{dx})(y)}{(x+7y)^2} = ...
0
votes
3answers
24 views

Fairly simple differentiation question

Ok so the question is : If $f(x) = \frac {e^x} {x^6}$ Find $f'(x)$. I'm fine finding the answer, I know $\frac {e^x}{x^6} = e^x * \frac {1}{x^6}$ so I went ahead and used the product rule and got ...
0
votes
1answer
40 views

Maxima and minima of partial derivatives

I'm currently on the topic of finding maxima/minima for partial derivatives. However, I've recently come across a question which is rather confusing. Given: $$f(x,y) = x^3 -y^2 + 3x for (x,y) R^2 ...
0
votes
1answer
76 views

The values of $k$ for which $ \log(2x) \leq kx \leq e^{x/2}$ for all $x > 0 $

So I'm trying to solve a system of equations and I checked some other guys solution and he divides the function by the derivate, like so: $f(x)/f'(x)$. Find the values of the real constant $k$ for ...
0
votes
0answers
29 views

Total derivative of scalar function with respect to a vector

If i have a real valuad scalar function $f(y(x),z(x))$ with $y(x): \mathbb{R}^{n_x} \mapsto \mathbb{R}^{n_y}$ and $z(x): \mathbb{R}^{n_x} \mapsto \mathbb{R}^{n_z}$ and i want to get the total ...
1
vote
2answers
73 views

How to differentiate $y=x^{y^{\sin x}}$

I know I'll have to use implicit differentiation, but I always get stuck when there is an exponent with trig, log, and/or natural log.
2
votes
4answers
41 views

Let $z = x^a y^b \ln(xy)$. Find $x \frac {dz} {dx} - y \frac {dz} {dy}$ in terms of $z$

I'm baffled by this question. I assume I'm meant to use the product rule to work out $\frac{dz}{dx}$ and $\frac{dz}{dy}$? But when I'm doing that I'm getting crazy answers that I know are wrong: ...
3
votes
3answers
124 views

How to evaluate $\lim\limits_{x\to 0} \frac{\sin x - x + x^3/6}{x^3}$

I'm unsure as to how to evaluate: $$\lim\limits_{x\to 0} \frac{\sin x - x + \frac{x^3}{6}}{x^3}$$ The $\lim\limits_{x\to 0}$ of both the numerator and denominator equal $0$. Taking the derivative ...
3
votes
5answers
721 views

Chain rule with triple composition

We are supposed to apply the chain rule on the following function $f$: $$ f(x) = \sqrt{x+\sqrt{2x+\sqrt{3x}}} $$ I assumed we could rewrite this as $$ f(x) = g(h(j(x))) $$ However, I was not sure ...
2
votes
1answer
79 views

Analysis-Baby Rudin's differentiability and continuity: theorem 5.2 and 5.6

I am very confused about differentiability and continuity. At the beginning of the differentiation chapter, we proved that differentiability contains continuity. (Theorem 5.2) But in example 5.6 and ...
1
vote
1answer
89 views

Definition of Point of Inflection

An inflection point is a point on a curve at which the sign of the curvature (i.e. the concavity) changes. According to Wikipedia, "If x is an inflection point for f then the second derivative, ...
1
vote
2answers
34 views

Approximate $f(1.01)$ for a function satisfying $f'(x) = 3f(x) + 3x$ and $f(1)=3$

Suppose that the derivative of a function satisfies the formula $f'(x) = 3f(x) + 3x$. If $f(1)=3$, use linear approximation to estimate the value of the function at $1.01$. I think I found $f(x) = ...
0
votes
2answers
34 views

Linear Approximation.

Use linear approximation to approximate the number $ln(1.02)$. This is what I did and it is still wrong on my online homework. $f(x) = ln(x)$ $f'(x) = \dfrac{1}{x}$ $y=\dfrac{1}{x}(x-1)$ ...
5
votes
1answer
66 views

Prove that if $f$ is differentiable at $x=0$, then $f$ is differentiable on $\mathbb{R}$.

$Conj:$ Suppose that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$, satisfies $f(a + b) = f(a)f(b)$ for all $a,b,\in\mathbb{R}$, and is not identically zero ($\exists ~x$ ...
4
votes
1answer
208 views

The proper and easiest way of doing an integral with derivative?

I have this integral: $$\int{\sec^3x\,\mathrm dx}$$ I don't understand how I would solve this. Google and YouTube videos don't help me understand much, other than just giving the answer. Is it ...
3
votes
1answer
37 views

How do I Implicitly Differentiate this equation?

My equation is $y=x^{y^2}$ I did the $\ln$ of both sides, then I tried implicit differentiation. I got $$y'= \frac{x^{y^2} y^2}{x}.$$
2
votes
0answers
44 views

Multivariable chain rule with vector valued function

Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $\mathbf{g}:\mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\mathbf{x} \in \mathbb{R}^n$. How do I find a formula for $\nabla f(\mathbf{g}(\mathbf{x}))$?
1
vote
1answer
25 views

Differentiate this equation (below):

$$\Large y = x^{\ln 7} + \log_7 x $$ I know for differentiating logarithms you do: $1/f(x) \cdot f'(x) \cdot 1/\ln b$. But how about differentiating $x^{\ln 7}$? I don't understand how to change ...
6
votes
3answers
100 views

Solve for constants: Derivatives using first principles

Question Find the values of the constants $a$ and $b$ such that $$\lim_{x \to 0}\frac{\sqrt[3]{ax + b}-2}{x} = \frac{5}{12}$$ My approach Using the definition of the derivative, $$f'(x) = ...
1
vote
1answer
34 views

evaluation of $\nabla \cdot ( \boldsymbol{B}(\boldsymbol{x}) \cdot \boldsymbol{B}^{T}(\boldsymbol{x}) )$

i have a symmetric positive matrix $\boldsymbol{D}(\boldsymbol{x})$ which can be decomposed as: $\boldsymbol{D}(\boldsymbol{x})$ = $\boldsymbol{B}(\boldsymbol{x}) ...
0
votes
1answer
37 views

Constructing Manifolds: Submersion

Given a smooth manifold $M$ and a topological space $N$. Consider a local homeomorphism $F:M\to N$ with $\mathrm{im} F=N$. Then one can turn the target space into a smooth manifold via: ...
1
vote
2answers
73 views

Concavity of function $F(x) = x^{1/5} (x+6)$

I was wondering when this function would curve upwards/downwards. I was having trouble finding the inflection points. Thank you. $$F(x) = x^{1/5} (x+6)$$ Progress I found the first derivative to be ...
2
votes
1answer
44 views

Functions of several variables and $Df$

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $g(x_1,...,x_n)=x_1^5+...+x_n^5$. Suppose $g\circ f\equiv 0$. Show that ...
1
vote
2answers
55 views

Use limit definition to find derivative of $x+\sqrt x$

The function is $f(x) = x + \sqrt x$. How would you use the limit definition of the derivative to find the derivative of that equation?
1
vote
1answer
37 views

Divergence of a vector field?

For some vector field $F = f(x)i + g(y)j$, the divergence in $\mathbb{R}^2$ is defined by: $\frac{\partial {f}}{\partial {x}} + \frac{\partial {g}}{\partial {y}}$. What happens if $f$ or $g$ is not ...
0
votes
6answers
70 views

If $f(x)\ge g(x)$, is $f'(x)\ge g'(x)$?

We choose any function for $f(x)$ and $g(x)$. Also, $x$ needs to be positive at all times. Lets say that $f(x)=45x^2$ and $g(x)=15x^2$. We can say that $f(x)\ge g(x)$, if $x\ge 0$. So the condition ...
17
votes
2answers
863 views

A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
0
votes
1answer
43 views

Stream functions and divergence?

We see that the existence of a stream function guarantees that the vector field has zero divergence or, equivalently, is source free. The converse is also true on simply connected regions of ...
2
votes
2answers
55 views

Solving $x\sin(\frac 1x)$ via limit definition

I'm trying to show that the derivative of $x\sin(\frac 1x)$ exists and is equal to $\sin(\frac 1x)-\frac {\cos(\frac 1x)}x$ for every point in its domain via the limit definition (I can of course just ...
1
vote
0answers
48 views

Computing Jacobian of error function using Lie Algebra

First off all, I hope this is the right place to ask, as it is a computer vision problem, but I'm specifically asking about the mathematical part of it. I am currently implementing the ICP (Iterative ...
4
votes
2answers
109 views

Computers can't deal with limit of $\Delta x \to 0$

While I was studying about finite differences I came across an article that says "computers can't deal with limit of $\Delta x \to 0$ " in finite differences.But if computers can't deal with these ...
0
votes
0answers
51 views

Critical Points of a Complex Sine Function within Bounds

I need a method to find the critical points of the function below. f(x) = 3.8*sin(2.4*x + 1) - 2.3*sin(7.2*x - 2) + 3.2*sin(8.1*x - 3) Bounds [-10, 10] I ...
0
votes
1answer
47 views

Complex Analysis - Complex plane, differentiable

Determine all the points in the complex plane where the function f(z) = tan(z) is differentiable and calculate the derivative at those points.
0
votes
0answers
36 views

Extreme-Value Theorem?

The extreme-value theorem states that if a function $f(x_{1},x_{2},...x_{n})$ is continuous in a closed and bounded interval within the domain of $f$, there exists both an absolute maximum and ...
0
votes
3answers
84 views

How can I determine a general formula for the nth derivative of any continuous function f(x) differentiable at least n times?

I know how to do it with easier functions, but is there a universal method which can be applied to all continuous functions differentiable at least n times(introduced to in a second year calculus ...
2
votes
1answer
54 views

Inflection point not found for the function $f(x) = 2\arctan(x) - \dfrac{x^3}{x^2+1}$. Should it?

$f'(x) = -\dfrac{x^4+x^2-2}{\left(x^2+1\right)^2} = \dfrac{(x+1)(x-1)(-x^2-2)}{\left(x^2+1\right)^2}$ This gives the critical points $x=-1 \quad\&\quad x=1$. Solving those with sign analysis; ...
2
votes
1answer
112 views

how to maximize weekly revenue using profit function and derivatives

p=45-0.01q where p is price of each product sold and q is the quantity of products sold. a) find the quantity that maximizes the weekly revenue of the company b) what price should the company sell ...
4
votes
1answer
60 views

Solve $f'(x) = 0$ and set up a sign chart for $f'$.

I understand how my teacher got the two $x$ values, but why didn't he solve for $e^x=0$? I know he did $x=0$ which is $0$ $x+2=0$ which is $-2$ so why no $e^x=0$? is there even an answer for ...
0
votes
1answer
53 views

Direction of unit vector that maximize directional derivative

Firstly, I am aware that there are quite a few question regarding with "maximizing direction derivative" already being asked. But after scanning through, I am still not able to figure out my question ...
3
votes
6answers
70 views

Derivative of $y=x^{\ln x}$?

I only know how to do one step: $$ \ln\left(\,y\,\right) = \ln\left(\, x^{\ln\left(\, x\,\right)}\,\right) $$ how do i do the derivative of $\ln\left(\, x^{\ln\left(\, x\,\right)}\,\right)$ ?. I know ...
4
votes
3answers
50 views

What is the derivative of $\arccos(x^2)$?

So I know that the derivative of arccos is: $-dx/\sqrt{1-x^2}$ So how would I find the derivative of $\arccos(x^2)$? What does the $-dx$ mean in the above formula? Would it just be ...
1
vote
1answer
26 views

Simplifying this Complex Radical: $-{1/5}(x-4)^{5/3} - 2(x-4)^{2/3}$

$-{1/5}(x-4)^{5/3} - 2(x-4)^{2/3}$ I need to get this function into a simpler form so that I can analyze it's domain, limits, derivative and second derivative more easily. I am very bad with radicals ...
1
vote
1answer
71 views

Why is $\frac{d^2}{dx^2} = \frac{1}{\delta^2}\frac{d^2}{dX^2}$ after the substitution $X = \frac{x}{\delta}$?

I have been working on this for a while and it makes no sense to me. Here's the deal. We're using the substitution $X = \dfrac{x}{\delta}.$ Then, for some reason, this gives us $$\begin{split} X = ...
0
votes
1answer
19 views

How do I solve this derivative given limited information?

Given $y=\cos^{-1}(t^{-1})-\sec^{-1}(t)$, I am to find $y'$. I am also given the following definitions: $\frac{d}{dx}\cos^{-1}(x)=\frac{-1}{\sqrt{1-x^2}}$ ...
0
votes
0answers
55 views

Multivariate derivatives

I have a simple problem im looking at and I dont understand some concepts. Let me define a multivariable polynomial $f(x_1,\ldots x_n,y_1,\ldots y_n)$. If I need to compute the value of the ...
0
votes
0answers
31 views

If a function is continuously derivable is it also continuously differentiable?

I'm doing an computer science online test and my professor is putting loads of trick questions in in. I'm wondering whether this is also one. I have a question for which we stated at the lectures ...
0
votes
1answer
41 views

An obstacle in a proof of Lagrange's mean value theorem by Nested Interval theorem

I was trying to prove Lagrange's mean value theorem by Nested Interval theorem and there's step where I got stuck ; let me write down to the step Let $f:[x_1,x_2]\to \mathbb R$ be continuous on ...
2
votes
3answers
37 views

Derivative of an exponential function

I am trying to solve $$\frac{1}{e^{x}}$$ I first tried using the quotient rule, and ended up with: $$\frac{e^{x}}{(e^{x})^2}$$ That was not the right answer, so I took a look at wolfram, and ...
1
vote
4answers
91 views

Find the derivative of y with respect to the given independent variable

Find the derivative of y with respect to the given independent variable: $y = 3^{-x} \stackrel{D}{\longrightarrow} y' = 3^{-x} \cdot (-1) \cdot \ln 3 $ This is my teacher's solution. I don't ...