What is Product rule: Definition and 139 Discussions
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as
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{\displaystyle (u\cdot v)'=u'\cdot v+u\cdot v'}
or in Leibniz's notation as
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{\displaystyle {\dfrac {d}{dx}}(u\cdot v)={\dfrac {du}{dx}}\cdot v+u\cdot {\dfrac {dv}{dx}}.}
The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts.
Hi,PF
The book is "Calculus" 7th ed, by Robert A. Adams and Christopher Essex. It is about an explained example of the first conclusion of the Fundamental Theorem of Calculus, at Chapter 5.5.
I will only quote the step I have doubt about:
Example 7 Find the derivatives of the following...
##\tiny{2.4.2}##
Differentiate ##f(x)=x\cos{x}+2\tan{x}##
Product Rule ##[-x\sin{x}+\cos{x}]+[2\sec^2]\implies \cos{x}-x\sin{x}+2\sec^2x##
mostly just seeing how posting here works
typos maybe
suggestions
what forum do I go to for tikz stuff
v=197
If $y=x \sin x,$ then $\dfrac{dy}{dx}=$
$a.\quad\sin{x}+\cos{x}$
$b.\quad\sin{x}+x\cos{x}$
$c.\quad\sin{x}+\cos{x}$
$d.\quad x(\sin{x}+\cos{x})$
$e.\quad x(\sin{x}-\cos{x})$
well just by looking at it because $dx(x) = 1$
elimanates all the options besides b
$1\cdot \sin (x)+\cos (x)x$...
I thought this was kind of a cool proof of the product rule.
Let ##F(x)## and ##G(x)## be cumulative distribution functions for independent random variables ##A## and ##B## respectively with probability density functions ##f(x)=F'(x)##, ##g(x)=G'(x)##. Consider the random variable...
MTW p 257, exercises 10.2 through 10.5: These exercises are all dealing with this familiar property of derivatives ∇ (AB) = ∇A B + A ∇ B . I learned this was called the "product rule". I learned that d/dx f(y(x)) = df/dy dy/dx is called the "chain rule". MTW keeps calling what I learned as the...
Homework Statement
This isn't really a homework problem, as the entire solution is laid out in the text. My question is in regards to a possible typo, which I have highlighted in blue in the given picture.
Usually I don't like to second guess the text, but this one has been absolutely plagued...
Homework Statement
I am facing problem in applying the chain rule.
The question which I am trying to solve is,
" Find the second derivative of "
Homework Equations
The Attempt at a Solution
So, differentiated it the first time,
[BY CHAIN RULE]
And now to find the second derivative I...
Homework Statement
Counting Internet Addresses In the Internet, which is made up of interconnected physical networks of computers, each computer (or more precisely, each network connection of a computer) is assigned an Internet address. In Version 4 of the Internet Protocol (IPv4), now in use...
I have a very basic knowledge of calculus of one variable .
In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given .
Then the book says, differentiating you get
PdV +VdP = nRdT .
The book doesn't explain the differentiation step .
I think , there are two ways to...
Homework Statement
find derivative of (x-2)(x-3)^2
Homework Equations
using product rule.
The Attempt at a Solution
1(x-3)^2+2(x-3)
x^2-6x-9 +2x-6
x^2-4x-15
doesn't factor.
It is well known that the product rule for the exterior derivative reads
d(a\wedge b)=(da)\wedge b +(-1)^p a\wedge (db),where a is a p-form.
In gauge theory we then introduce the exterior covariant derivative D=d+A\wedge. What is then D(a ∧ b) and how do you prove it?
I obtain
D(a\wedge...
What is the correct way to write the product rule in Newton notation (with the dots above)? It is the LHS I am abit confused with. Eg. Say you have d/dt(xy) would you just put dots above the x and y?
Hi! I was reading the Wikipedia article on Newton's laws of motion. I read there that when mass is a variable function of time as well as velocity, one cannot use the product rule of derivatives to expand d/dt(mv)
It said that d/dt(mv)=mdv/dt+vdm/dt is WRONG
I don't know why that is wrong. The...
Homework Statement
Verify the identity:
## \nabla \times ( A \times B) = (B\bullet \nabla)A - (A\bullet\nabla)B + A(\nabla \bullet B)-B(\nabla\bullet A)##
My issue here is I don't understand the significance of why a term has B or A on the left of the dot product, and another has B or A on...
Homework Statement
Homework Equations
The product rule formula.
The Attempt at a Solution
I managed to solve 45/50 product rule but I can't seem to solve these ones. Apparently you use product rule to solve these.
Homework Statement
Basically, I'm looking at the property that says if the magnitude of a vector valued function is constant, then the vector function dotted with it's derivative will be zero. But I'm stuck towards the end because the proof I found online seems to skip a step that I'm not...
Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused
I'm trying to understand how the derivative of this function:
x=ρcosθ
Becomes this:
dx=−ρsinθdθ+cosθdρ
First off I'm guessing that x is a function of both ρ AND cosθ, or else we wouldn't be using the product rule in the first place..Am I correct? So how could we write this in functional...
Im stuck on theorem 5 where the book used chain rule then used product rule then again using the chain rule. How in the world does it work? I don't get product rule used and chain rule used after.
Homework Statement
Prove that $$log_{b}(xy)=log_{b}x+log_{b}y.$$
Homework Equations
Let $$b^{u}=x,b^{v}=y.$$ Then $$log_{b}x=u,log_{b}y=v.$$
The Attempt at a Solution
I'm afraid I've been using circular reasoning to prove this. I can get this to a point where I have...
Say I have a position vector
p = e(t) p(t)
Where, in 2D, e(t) = (e1(t), e2(t)) and p(t) = (p1(t), p2(t))T
And if I conveniently point the FIRST base vector of the frame at the particle, I can use: p(t) = (r1(t), 0)T
I want the velocity, so I take
v = d(e(t))/dt p(t) + e(t) d(p(t))/dt...
www.youtube.com/watch?v=oW4jM0smS_E
That's the video I'm referencing in particular, but 1 and 3 are necessary prereqs if you're new to the matter (as I am).
He goes through and derives the product rule and power rule for polynomials using algebra.
My question is this: why don't we teach...
What has done here in the second line of the proof for product rule?, from Mathematical methods for physicists from Riley, Hobson
they defined f(x)=u(x)v(x) and these steps are given,
I have no idea how to proceed further please help me.
Homework Statement
[/B]
hi could some body please help me factorise this please ? any chance of a few stages would be much appreciated
Homework EquationsThe Attempt at a Solution
my attempt , but my solutions say otherwise ?
[/B]
Hello,
I have this exercise that I can't get the right answer. I have to find derivative of
g(x)= (4${x}^{2}$-2x+1)${e}^{x}$
So, what is did is
g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$
My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did...
Homework Statement
Saw a calculation that put differentiation of power in terms of acceleration as follows:
E=Fs
dE/dt=Fv=P
dP/dt=Fa=ma^2
It doesn't make sense to me because if power was changing, acceleration must change. Correct me if I'm wrong, but shouldn't the product rule be applied...
when differentiating
e^(at) * (cos(bt) + isin(bt))
are you able to use product rule to find the derivative considering (cos(bt) + sin(bt)) as one function??
why??
and what does d/dt exactly mean?? (they get multiplied to a function that needs to be differentiated and I wanted to...
1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s
2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)
3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to...
Definition/Summary
The product rule is a method for finding the derivative of a product of functions.
Equations
(fg)'\ =\ f'g\ +\ fg'
(fgh)'\ =\ f'gh\ +\ fg'h\ +\ fgh'
Extended explanation
If a function F is the product of two other functions f and g (i.e. F(x) = f(x)g(x))...
Hey guys, just trying to understand how the quotient rule is derived, so I head over to wikipedia and saw this:
But I'm having some difficulty understanding what goes on between these two steps:
Could someone shed some light on this?
Can someone check my working. I don't understand why i am getting different answers?
u(x,t)=\frac{{e}^{-\frac{x^2}{4Dt}}}{\sqrt{4Dt}}
Differentiate w.r.t 't' by quotient rule:
\frac{\partial u}{\partial t}=\left[ \frac{1}{\sqrt{4Dt}}\cdot \frac{x^2}{4Dt^2}\cdot...
I was working on a pde, and I needed to compute a Jacobian for it.
Suppose we have a function consisting of a series of matrices multiplied by a vector:
f(X) = A * B * b
--where X is a vector containing elements that are contained within A, b, and/or b,
--A is a matrix, B is a matrix, and b is...
I was working on PDE for a project and needed to compute a Jacobian for it.
Suppose we have a function consisting of a series of matrices multiplied by a vector:
f(X) = A * B * b
--where X is a vector containing elements that are contained within A, b, and/or b,
--A is a matrix, B is a...
Homework Statement
Find the gradient of the curve at the given point on the curve
y = \frac{(√x - 1)}{√x} where x = 9
Homework Equations
y(x) = u(x)v(x)
dy/dx = u(dv/dx) + v(du/dx)
The Attempt at a Solution
my problem really boils down to rearranging the function to a form...
I have always wondered:
Is the product rule and addition rule for that matter axioms of the probability theory or can they actually be proven from more general principles? The reason I ask is, and it might be a bit silly, that I have always thought I missed out on something in probability...
Hello all,
I'm having trouble with proving that the derivative of f(x)*g(x) is f'(x)*g(x)+f(x)*g'(x).
Now, I've already seen the actual proof, and I can understand its reasoning, but the first time I tried to prove without looking at the solution, this is what I wrote before I became rather...
Homework Statement
The question I am trying to answer requires me to find the following:
dN/dS ∝ S^−5/2/cosh(r/R)
and I am giving the follwing equation in the question.
A=4πR^2 sinh^2〖(r/R)〗
The Attempt at a Solution
Right I know how to get the S^-5/2 in the top half of the...
Hello, first post here.
I am preparing for my Introductory Quantum Mechanics course, and in the exam questions, we are asked to use Ehrenfest's theorem to show that
\frac{d}{dt}\langle \vec{r}\cdot \vec{p} \rangle = \langle 2T-\vec{r}\cdot \nabla V \rangle
Now, from other results...
Is the rule:
P(AB I) = P(BA I)
(which is used to derive Bayes rule) an axiom for probability? And if so, do you guys find it intuitive that it should hold. For instance consider a box with green and red beads. Do you think it is strictly obvious that the probability of getting red-green is...
Hello MHB,
I got one exempel that I don't get same result as my book.
Exempel: If z=f(x,y) has continuos second-order partial derivates and x=r^2+s^2 and y=2rs find \frac{d^2z}{dr^2}
So what I did before checking soulotion:
\frac{d^2z}{dr^2}=\frac{dz}{dr} \frac{d}{dr}
So I start with solving...
Homework Statement
Find the integral of
This was the question. There is a way to do it by long division but I am confused with Long division. Instead I tried to do by the method below but I failed...
Homework Equations
NoneThe Attempt at a Solution
I thought maybe I could reduce the...
Homework Statement
(1-2x3+x2)((1/x3)+1)
Homework Equations
f'g(x)+g'f(x)
f'(x)=-6x3+2x
g'(x)=(-1/x4)
The Attempt at a Solution
I found what i believe to be the derivative of both f(x) & g(x) and used the product rule to get to where I am stuck right now:
[(-6x2+2x)(1/x3)]+[(-1/x4)(1-2x3+x2)
Homework Statement
If f(2) = 3, f'(2) = 5, g(2) = -1, g'(2) = -4, find (fg)'(2).
Homework Equations
if F(x) = f(x)g(x)
F'(x) = f'(x)g(x) + g'(x)f(x)
The Attempt at a Solution
I have no idea how to attempt his question :(