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

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Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
3
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3answers
33 views

How to find the derivative of $(3x-1)^2(2x+3)^2$

I used the power rule and the chain rule and ended up with this: $$y'= (3x-1)^2 \times 2(2x+3) \times 2 + (2x+3)^2 \times 2(3x-1)\times 3$$ The next step, which I do not understand how it is combined ...
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4answers
34 views

derivative of $f(x)=(4x^2+9)^7(7x^2+3)^{12}$

$f(x)=(4x^2+9)^7(7x^2+3)^{12}$ I used the product rule and came up with: $y'=4(x^2+9)^7(168(7x^2+3)^{11})+56x(4x^2+9)^6(7x^2+3)^{12}$ Why is this wrong?
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4answers
56 views

Derivative of $f(x)=-6\sin^4 x$

$f(x)=-6\sin^4x$ $f(x)=-6\sin x^4$ $f'(x)=-6\cos x^4(4x^3)=-24x^3\cos x^4$ What am I doing wrong? Please show the steps.
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1answer
27 views

Derivative Functions [on hold]

Consider $f(x)= ax^2 + bx$ where $a$ and $b$ are real numbers. If $f(1)=-1$ and $f'(-1)=-7$, find the values of $a$ and $b$? I genuinely do not understand how to do it! Please help
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1answer
15 views

Differentiation of the following:

Given $h(x) = \frac{f(x)}{g(x)}$, find $h'(3)$ $f(1)=2, f'(1)=-1, g(1)=1, g'(1)=2 $ $f(2)=1, f'(2)=0.5, g(2)=3, g'(2)=0 $ $f(3)=3, f'(3)=2, g(3)=1, g'(3)=-2$ What I am getting is $-7/2$ , but I am ...
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3answers
42 views

Proof of the fact that ln(a) = f '(0) for f(x) = a^x?

Looking over notes from class today and wanted to know if there is any type of proof for the fact that $\ln(a) = \lim_{h\to0}(a^h-1)/h$, which is just $f '(0)$ for any function of the form $f(x) = ...
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0answers
16 views

Finding the Schwarzian Derivative

Determine the Schwarzian derivative of the following function: $f(x)=ax^2+bx+c$. I've plugged this into the Schawrzian derivative equation and got the following answer, but I'm not sure if it's able ...
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1answer
15 views

Stationary Points and Their Nature [on hold]

Use the Second derivative to find the stationary values and their nature. y=sin1/2θ + sin2θ.
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1answer
24 views

Derivatives without simplifing

Determine the derivative of the following function (do not simplify your answer): $\dfrac{4x^2-8x+14}{x^7-6x^3+4x-9} $ Do I use the quotient rule?
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0answers
10 views

Is it possible for the derivative of a multivariate function to be a function of lesser dimension?

Let's say I have some function $f$ such that $f'(a,b,c,d)$ exists for all $a$, $b$, $c$, and $d$, and that $f(a,b,c,d)$ is dependent upon $a$, $b$, $c$, and $d$. (That is, $f(a,b,c,d)$ can't be ...
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2answers
26 views

Derivative Help for Calculus

Determine the derivative for the following function (Do not simplify your answer):$f(x)= ( x^2+3x−5)^7 (5x^3+4x^2−3x+8) $. So far I got $ (2x+3x-5)^7 (15x^2+8x-3x+8)$ What would I do after this step?
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1answer
21 views

Misconception about chain rule in multiple variables

Let $z=f(x,y)=e^{x}\sin(xy)$, $x=g(s,t)$ and $y=h(s,t)$. If $k(s,t)=f(g(s,t),h(s,t))$, find $\displaystyle\frac{\partial k}{\partial s}$. Until now, I have found that: $\displaystyle\frac{\partial ...
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2answers
29 views

Differentiability of piecewise function using the definition analytically

In the following example particularly $$ f(x)= \begin{cases} (x-2)^2 + 5 \quad\text{when $x\geq 2$} \\ (x-2)^2 + 4 \quad\text{when $x<2$} \end{cases} $$ for the above function I know quite ...
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2answers
51 views

What's the difference between $\frac{\partial f}{\partial x}(x,y)$ and $\frac{\partial f (x,y)}{\partial x}$?

Is there a difference in the meaning of $\frac{\partial f}{\partial x}(x,y)$ and$\frac{\partial f (x,y)}{\partial x}$ or is it just a different notation for exactly the same thing?
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1answer
34 views

How to determine the differentiability of a piece-wise defined function?

How should one test for differentiability of a function that is piecewise defined? How is the differentiability related to one sided derivative in this case? Also, in the limits $\lim_{h\to 0^+}$ ...
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0answers
13 views

Show that if f is strongly differentiable at $x_0$ then it satisfies Lipschitz condition in a neighbourhood of $x_0$.

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that $f$ is strongly differentiable ...
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1answer
30 views

Show that $\frac{\mathrm{d^{2}}B }{\mathrm{d} A^{2}}> 0 $ if $U''<0$.

Given, $A = W_0 - L_0 + I - qI$, $B = W_0 - qI$, and $EU = p U(A) + (1-p) U(B) = k$, where $k$ is a constant. $\frac{\mathrm{d} B}{\mathrm{d} A}\bigg|_{}^{EU=k} = \frac{\frac{\partial EU }{\partial ...
3
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1answer
183 views

Calculus integral evaluation using substitution

I have to find this integral: Evaluate the integral using an appropriate substitution $$\int\dfrac{8e^x+7e^{-x}}{8e^x-7e^{-x}}\mathrm dx.$$ I've tried my solution $\ln\Big[15\cdot \sinh(x) + ...
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2answers
19 views

Derivative of $-\csc(2x)$

Is the following reasoning correct? $$-\csc2x = -1[\csc(2x)\cdot\cot(2x)] = -1[\csc(2x)\cdot\cot(2x)\cdot2\cdot2]$$ I am unsure whether that is correct so far because I do not know if the derivatives ...
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2answers
26 views

Derivative of $3 \cdot sin(2x)$

Is the derivative of $3 \cdot sin(2x)$ $3 \cdot [cos(2x) \cdot 2]$ or $3\cdot [cos(2x)]\cdot 2$? I'm unsure about this technicality. The first one seems more reasonable to me.
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4answers
12 views

Differentiability question ends up in contradiction.

Let $f(x)=x^3cos\frac{1}{x}$ when $x\neq0$ and $f(0)=0$. Is $f(x)$ differentiable at $x=0$? My first attempt Definition: A function is differentiable at $a$ if $f'(a)$ exists. $$f'(x)=\lim_{h ...
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2answers
19 views

When to use the chain rule

Would I use the chain rule in the following derivative problem: $$(sinx/x)$$ So far I have simplified it to: sin(x)(-1x^(-2))+x^(-1)(cosx) Would I have to further take the derivative of cosx ...
2
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1answer
51 views

Differentiate $e^x+x^e$

My answer was $e^x+ex^{e-1}$. As I understand it, the derivative of $e^x $ is $e^x$. As for $x^e$, I made this $e(x^{e-1})$ and simplified from there. Where was my error?
0
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1answer
25 views

Related rates: Rocket launch. dy/dt = 0.8, y = 5, x =?

There is a vertical rocketlaunch. x represents horizontal distance y represents vertical distance Right after launch: y = 5 ; dy/dt = 0.8 Given: Distance between launch pad and radarstation: x1 = ...
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0answers
17 views

differentiation of log of a sum of vectors

I would very much like to be able to differentiate the following function with respect to $x$: $$ln(\sum \limits_{j=1}^{d} \theta_j \bf a_j^2)$$ Where $\bf a_j$ is a $d \times 1$ vector with $x$ in ...
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0answers
20 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
0
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1answer
20 views

Use Leibniz' formula to show that the $(2n)$th derivative of $(2x^2 + 3x +1)sinx$ is $(-1)^n(2x^2+3x-8n^2+4n+1)sinx+(-1)^{n+1}(8nx+6n)cosx$ wrt $x$

If I let $f=f(x)=sinx$ and $g=g(x)=2x^2+3x+1$ and $D=$ First derivative wrt $x$, $D^2=$ Second derivative wrt $x$ and $D^n=$ $nth$ derivative wrt $x$ then, Leibniz' formula states that $\displaystyle ...
2
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1answer
20 views

Total Time Taken Question

Distance of chord = Time taken to "swim" to the desalination plant = I'm stuck here! The textbook working out is as such: I don't understand how they have the 'k' or 1/2 the runs river at ...
0
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1answer
23 views

Shortest Chord from origin to function

worked solution: Is this found using the distance of a line equation, where instead of co-ordinate points they use functions, so the two functions are g(x) and x (because the origin is on the ...
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1answer
16 views

Finding implicit differentiation using power rule.

Original Function: $(\sin(\pi x)+\cos(\pi y))^2 = 2$ Step1: $2(\pi \cos(\pi x) - \pi \sin(\pi y)\frac{dy}{dx})(\sin(\pi x)+\cos(\pi y))$ Step2: $(2\pi \cos(\pi x)-2\pi \sin(\pi ...
0
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2answers
62 views

Integrate $\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$ [on hold]

Evaluate$$\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$$ Solution through either contour integral method or indefinite integral method please!
2
votes
3answers
91 views

Problem with roots

I am having a few problems with roots. This is apart of a larger question where I am taking the derivative of of a function. I know I got the first part right (answer key) but when I plug in root 2 ...
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3answers
21 views

Using product rule or chain rule

For the equation $y=e^xx-2e^x$, I used product rule for the left ($e^xx+e^x)$ and right ($2e^x$) and combined the two to get $y'=e^xx-e^x$. But when I tried to factor out $e^x$ in the original ...
2
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1answer
23 views

Finding the equation of a line perpendicular to a curve at a given point.

Find the equation of the perpendicular line of $y=e^{-2x^2}$ at the point where $x=1$. I found the derivative: $y'=e^{-x^2}-2x$. And when I plug in one to the derivative I get: $m=\frac{1}{e}-2$. I ...
0
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3answers
41 views

What is the difference between a Limit and Derivative?

When you get the derivative of a point, arn't you just getting the limit at that point? I'm not quite sure why they need to be named differently when they seem to be doing the same thing.
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1answer
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Find the domain of derivative of the function $f(x)=\mid \sin^{-1}(2x^2-1)\mid$.

Find the domain of derivative of the function $f(x)=\mid \sin^{-1}(2x^2-1)\mid$. I was a little confused about the modulus. I can do the derivative and even calculate the domain, but, the modulus ...
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1answer
47 views

Transform the following Differential Equation

Transform the differential equation: \begin{equation} x^2\frac{\partial{z}}{\partial{x}} + y^2\frac{\partial{z}}{\partial{y}}=0 \end{equation} Taking as new variables $u=x$, $v=1/y-1/x$, ...
4
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2answers
78 views

What intuition stands behind implicit differentiation

I'm trying to undestand implicit differentation Let's take as a an example equation y^2 + x^2 = 1 1. How i think about how the equation works I think the function as : if x changes then the y term ...
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1answer
23 views

Show that $f'_+(a)=f'(a+)$ if both quantities exist.

Show that $f'_+(a)=f'(a+)$ if both quantities exist. I'm not really sure where to start, any help is appreciated. I came up with this: If $f'(a^+)$ exists, then by definition $f'(a+) = \lim_{x\to ...
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2answers
46 views

Intermediate step for integration?

Among the various method of Integration there is one specific method(it may vary according to the terms) where for instance if we have a function as:$$\frac{px+q}{ax^2+bx+c}$$ To integrate this we ...
2
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3answers
40 views

Trigonometric Identity for tan?

Can somebody please help with this (probably simple) query. Given $$dx = R\,d(\tan\theta)$$ this can be expressed as $$dx = R\,\sec^2\theta\,d\theta$$ I can't determine where the ...
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1answer
45 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
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1answer
47 views

A question on Lie derivative

For a Lie derivative $\mathscr{L}_{X} Y$ of $Y$ with respect to $X$, we mean that for two smooth vector fields $X$ and $Y$ on a smooth manifold $M$ such that the following holds $$ \mathscr{L}_{X} Y = ...
2
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0answers
34 views

Differentiation - a technical point

I understand the following equation to be correct, but why can we treat the differentials as fractions and cancel them out? What would be the correct way to view it? $$ \int_{-\pi/a} ...
0
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3answers
27 views

Is this derivative correct?

I'm newbie at Calculus, so I'm doing some exercises of derivates, I know by the formula: $f(x) = \sqrt u$ $\frac {df(x)}{dx} = \frac{u'}{2 \sqrt u}$ that the derivate of the next function is: ...
2
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1answer
18 views

If all directional derivatives are $0$, the function is constant.

Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be differentiable in every point of the disc $ B_{r}(\vec{a})$. If $D_{\vec{y}}f(\vec{x})=0$ for $n$ linearly independent vectors $\vec{y}_{1}, ...
2
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1answer
20 views

A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, ...
5
votes
2answers
68 views

nth derivative of $e^{-x}\sin(x)$

I'm trying but no luck. Can't find a pattern yet. The exercise is to find the nth derivative of $e^{-x}\sin(x)$ probably by induction.
1
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1answer
29 views

Proving convexity of a function whose Hessian is positive semidefinite over a convex set

C is a convex set in R^n and f:R^n --> R is twice continuously differentiable over C. The Hessian of f is positive semidefinite over C, and I want to show that f is therefore a convex function. I ...