1
vote
2answers
21 views

Evaluating $(\frac{\cos x}{1-\sin x})^2$

$(\dfrac{\cos x}{1-\sin x})^2$ $f\;'(x)= 2(\dfrac{\cos x}{1-\sin x}) \times (\dfrac{-\sin x+\sin^2x-\cos^2x}{(1-\sin x)^2})$ Does $\sin^2x-\cos^2=1$? or $-1$? Then it could factor with the ...
1
vote
2answers
29 views

How to find the derivative of $f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$?

Find the derivative of the following: $$f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$$ Would I use the chain rule and product rule? So far I have: $$\begin{align}g(x)=x^3-4x+6 \\g'(x)=2x^2-4\end{align}$$ would ...
0
votes
2answers
31 views

Derivative Help: $f(x) = x^3\,e^{5x-7}$

I need to find the derivative of the following function: $${\rm f}\left(\,x\,\right)= x^{3}{\rm e}^{5x - 7}$$ but I don't know where to start with this problem. Please help.
0
votes
1answer
31 views

Find the equation of normal line to the graph $y=2(x-1)^3$

Find the equation of normal line to the graph $y=2(x-1)^3$ at the point where $x=\frac12$. So far, I found the derivative: $$\frac{dy}{dx}= 6(x-1)^2 $$ What to do next?
2
votes
1answer
36 views

This is true ? This limit is a problem for me :/

Let $\phi:\mathbb{R} \to \mathbb{R}^n$ and $\lim_{t \to \infty} \phi(t) = X_0$, where $X_0$ is a constant in $\mathbb{R}^n$ then $\lim_{t\to \infty} \phi'(t) = 0$. I search everywhere and I need to ...
1
vote
3answers
28 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
2
votes
3answers
36 views

evaluating derivative of $\log_4(2x^2+1)$

Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ $\log_4(2x^2+1)=y$ $4^y=2x^2+1$ $4^y\ln4 \times y\;'=4x$ $y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ What ...
1
vote
4answers
26 views

Evaluating $\frac{d}{dx}\sqrt[4]{\ln(12-x^2)}$

Find Derivative and evaluate at $x=1$: $$ \frac{d}{dx}\sqrt[4]{\ln(12-x^2)} = (\ln u)^{1/4} $$ $$v=(v)^{1/4} \implies v=\ln\;u, v\;'=\dfrac{1}{u}(u\;')$$ $$y\;'=\frac{1}{4}v^{-3/4}\; \times ...
-1
votes
1answer
22 views

How to find equations of tangent lines to the graph passing through a line

How to find equations of tangent lines to the graph $f(x)=x/(x-1)$ passing through point $(-1,5)$? Progress I used the quotient rule and got $f'(x)=-1/(x-1)^2$, but I have no idea how to continue.
0
votes
1answer
20 views

Is this function of 2 variables differentiable?

$f(x,y) = \frac{\sin(x^4+y^4)}{x^2+y^2}$ when $(x,y) \neq (0,0)$ and $0$ when $(x,y) = (0,0)$ Is f differentiable?
-2
votes
1answer
25 views

How to use quotient rule to differentiate $f(t)=\frac{\cos t}{t^3}$? [on hold]

The function is $\displaystyle f(t)=\frac{\cos(t)}{t^3}$, and I want to know how to differentiate it using the quotient rule. Thank you so much!
1
vote
2answers
21 views

Chain rule for multiple variables?

What I've tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial ...
1
vote
4answers
53 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!
0
votes
1answer
21 views

Geometric interpretation of derivative?

For some function $F(x,y) = 0$, $$\frac{dy}{dx} = \frac{-F_x}{F_y}$$ Can someone give me a geometric interpretatio of this? ($F_x$ and $F_y$ are the partial derivatives)
0
votes
3answers
19 views

How to find the points where the slope of the tangent is $-1$?

For the function $f(x) = x^3 - 4x$, find the points where the slope of the tangent is $-1$. Use the algebraic method. Do I need just to find zeros?
0
votes
1answer
30 views

Second derivative of $\sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)}$

How to take second derivative of $$\sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)}.$$ I am having trouble with taking the second derivative of this. I know I should simplify it before taking the ...
0
votes
3answers
33 views

Evaluating $\frac{\operatorname d \! \phantom x}{\operatorname d\!x}\frac{4}{\ln(x^2+2)}$

$\dfrac{\operatorname d \! \phantom x}{\operatorname d\!x}\dfrac{4}{\ln(x^2+2)}= \dfrac{4}{\ln u}$ $u=x^2+2$ $u\;'=2x$ $y\;'=\dfrac{4}{\dfrac{1}{u}} \times (u\;') \implies ...
0
votes
4answers
36 views

Using parametric differentiation for $\frac{\operatorname d \! y}{\operatorname d \!x}$?

Hi so I'm in my calculus class and the teacher gave us a problem to do. I'm not quite sure how to attack this question. He's given us a couple of steps but I don't understand. If someone can further ...
0
votes
3answers
21 views

Derivative of $\frac{d}{dt}\ln(6t^2+9t+12)=$

$\dfrac{d}{dt}\ln(6t^2+9t+12)=$ $y=2\ln(6t)+\ln(9t)+\ln(12)$ $y\;'=2\dfrac{1}{6t}(6)+\dfrac{1}{9t}(9)+0$ $=\dfrac{12}{6t}+\dfrac{9}{9t}=\dfrac{2}{t}+\dfrac{1}{t}$ What am I doing wrong?
-1
votes
1answer
16 views

Finding the max with velocity and acceleration graph

I'm confused on why there is a maximum at R. If I flipped the acceleration graph it looks like a continuously increasing function with no max or min to me. Could someone help me understand this? 8a ...
0
votes
1answer
19 views

Implicit Differentiation of $\cos^3(y)$

I think I understand implicit differentiation outside of those problems involving trig functions but for some reason this problem is breaking my brain: Assume that y is a function of x. Find ...
1
vote
0answers
72 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
0
votes
1answer
27 views

Calc I limit/series question

Let $f : \mathbb R\rightarrow\mathbb R$ be a function that is differentiable at zero and such that $f(0)=0$. Show that for each $n\in \mathbb N$ we have that ...
1
vote
1answer
16 views

Calculus Question ( Surface area/Volume)

An open box (I.e. no lid) had a square base of side $x$ cm and height $h$ cm. Given that the volume of box is $108$ cm$^3$. a) Show that the surface area in cm$^2$ is given by $A = x^2 + 432/x$. b) ...
2
votes
1answer
17 views

Incongruencies with derivatives and differencials

I read in Piskunov that the increment $\Delta y$ of a function can be written as: $\Delta y = f'(x) \Delta x + \alpha \Delta x$ And, when ${\Delta x\to 0}$ , $dy=f'(x)dx$ The problem is, doesn't ...
0
votes
2answers
20 views

Find the equation of the line tangent to the curve $y=x^2$ parallel to the line $y=x$

Find the equation of the line tangent to the curve $y=x^2$ parallel to the line $y=x$. Just started A level maths, any help is appreciated.
1
vote
0answers
89 views
+200

Infinite Series -: $\psi(s)=\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. (Note the notional ...
0
votes
2answers
25 views

the derivative of cos(2x) with the double-angle formula?

So, last minute my teacher posted something saying to study double-angle formulas for our derivative test tomorrow. So in the back of the book it shows three things for $\cos x$ $2 \cos^2 x$ ...
0
votes
1answer
36 views

Derivative of all real x

Find the derivative of the function for all real $x$. $f(x)= (\sin(x^\frac 13)^3$) It also gives a hint saying extra attention needs to be placed on $x = 0$. Getting the basic derivative isn't the ...
1
vote
2answers
38 views

Computing the nth-derivative $\frac{d^{n}}{d\lambda^{n}}e^{\lambda x-\frac{\lambda^{2}}{2}t}$

According to wolfram-alpha, $\frac{d^{n}}{d\lambda^{n}}e^{\lambda x-\frac{\lambda^{2}}{2}t}= \frac{(-i)^{n} (-t)^{\frac{n}{2}} }{2^{\frac{n}{2}} }e^{x \lambda-\frac{t \lambda^2}{2}} H_n(\frac{(x-t ...
0
votes
2answers
46 views

Calculate the derivative of $\sqrt{1+\cot^2(x)}$

$$f(x) = \sqrt{1+\cot^2(x)}$$ How to calculate the derivative $f'(x)$? I've been looking at similar problems in my book and at examples, but I'm having a lot of trouble understanding it still. I'd ...
0
votes
1answer
11 views

Analyze the graph of a derivative

I have the graph of the derivative of some function: And i need to know: a) The critic values of f. b) The X coordinate of each of points where theres an relative extrema of f. c) An interval of ...
0
votes
3answers
44 views

How to find $\frac {dy}{dx}$ at the point

If $y=\dfrac{3x^2}{1-4x}$ I am solving through u/v formula but its not working for me Find $\dfrac{dy}{dx}$ for $x=1$.
0
votes
1answer
44 views

Differentials (Higher order)

I am having trouble thinking about $\text{dy}$ and $\text{dx}$, $\text{d}^2\text{y}$ and $\text{dx}^2$ as differentials. I get that you can write $\text{dy}=f'(x)\text{dx}$ and how to derive it but ...
0
votes
1answer
57 views

“The derivative of a sum is the sum of a derivative”. What?

According to this video at this time: "We're gonna do the chain rule here where the derivative of a sum is the sum of a derivative..." Can anyone explain to me why $$ \frac{\partial}{\partial ...
2
votes
3answers
51 views

Is $x=0$ an inflection point?

Consider $f(x)=x^{\frac {5}{7}}$, is it $x=0$ an inflection point? $$f'(x)=\frac {5}{7}x^{\frac {-2}{7}}$$ $$f''(x)=\frac {-10}{49}x^{\frac {-9}{7}}$$ As far as I know, the inflection point is the ...
1
vote
1answer
30 views

Finding Critical Values of Function

$$f(x)=x^{\frac{5}{11}}\cdot(x-5)^2$$ So far, I have used the product rule and chain rule to get... $$\left(\frac{5}{11}x^{\frac{-6}{11}}\cdot(x-5)^2\right)+\left(x^{\frac{5}{11}}\cdot(2(x-5))\cdot ...
0
votes
3answers
44 views

How to find this derivative using difference quotient?

how would i find the derivative of $x^8+12x^5-4x^4+10x^3-6x+5$? I know the answer is $8x^7+60x^4-16x^3+30x^2-6$. but how should i solve it using difference quotient, can someone please show the step ...
0
votes
2answers
39 views

Differentiate: $f(\theta) = \frac{\sec \theta} {3 + \sec \theta}$

I got $$\frac{(3+\sec \theta) (\cos\theta) - (\sec \theta) (3+\cos \theta)} { (3+\sec\theta)^2}$$ However the program that I am using says my answer is wrong.
-1
votes
2answers
36 views

Find the critical numbers for this function

$$f(x)=\frac{5x+7}{x^2+x+1}$$ I found the derivative of the function, which is $$\frac{-5x^2-14x-2}{(x^2+x+1)^2}$$ The problem is that I am not getting the correct values. Apparently the answer is ...
0
votes
1answer
21 views

Using differentials with volume of a cube

my question is The volume of a cube is increased from 1000 cubic centimeters to 1156 cubic centimeters. Use differentials to determine. the side length of the cube increases by? the surface area ...
5
votes
6answers
565 views

Problem in the second-derivative symbol.

The second derivative of this symbol according to the rules that we have learned the correct mathematical, I wish to know why this symbol is not used.
1
vote
5answers
55 views

Differentiate the following function: $y = \frac{2x^2 + 6\sqrt{x} }{9x}$

My answer is $- \dfrac{1}{9 \sqrt{x} }$, however, the program I am using states that I am wrong. Where have I went wrong?
0
votes
3answers
40 views

what is the derivative of $3\cos(\cos x)\;?$

what is the derivative of $3\cos(\cos x)\;?$ I think I need to use the chain rule and i believed it to be $3-\sin(\cos x)(-\sin x)$ but this is not the case.
0
votes
4answers
57 views

What's pratical use of Derivate function calculus? [duplicate]

I would like to know whats the pratical use of derivate calculus? Or what it means? If you can give some pratical example I'll be grateful. Eg.: I can use an definite integral to know area of a ...
0
votes
1answer
24 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
2
votes
2answers
84 views

Log base 10 not equal to log while differentiating?

I was looking at sample questions from my textbook and I came across something interesting that I need a little help understanding The question was to find the derivative of: $\log_{10} ...
0
votes
1answer
25 views

Term for functions with infinite derivatives [closed]

Functions that include a negative indice such as x-1 or similar have an unlimited number of derivatives, so f'(x), f''(x), and fn(x) exist. Is there a technical term for functions like these? I've ...
0
votes
0answers
27 views

Directional derivatives in two directions

How can I take a directional derivative in two directions? I mean,$$D_{xy}f(0,0)$$ Because when I have something like $$D_{x}f(0,0),$$ I just use that my direction is in the x axis, $ \vec ...
4
votes
1answer
48 views

Given the following derivatives, find the integrals

Find the derivatives of $\ln(x+\sqrt{x^2+1})$ and $\arcsin(x)$, and use the result to find the integrals of the following functions: $$ \dfrac{1}{ \sqrt{ \pm x^2 \pm a^2 }} $$ $$ ...