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
d
d
x
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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, I got the right answer when I used the Quotient Rule but not when I used the Product Rule...
I think it might be an algebra mistake...
Product Rule Method:
f'(x) = (3 - x^2)*(4 + x^2)^-1
= (3 - x^2)[(-1(4 + x^2)^-2)*2x] + [(4 + x^2)^-1](-2x)
= [(3 - x^2)(-2x)]/[(4 + x^2)^2] +...
Homework Statement
the product rule fn->f , gn->g implies fngn->fg true in the normed
vector space (C[0,1],||.||) depends on the the norm||.||. Give a proof or a
counterexample for (C[0,1],||.||infinite),(C[0,1].||.||1)Homework Equations
counterexample , you may wish to examine the case f=g=0...
Homework Statement
Use the product rule to differentiate the function:
h(t) = √t(1-t2)
Homework Equations
d/dx[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)
The Attempt at a Solution
(see attachment image)
I checked the back of the textbook and my solution was wrong. The textbook says...
Homework Statement
Find the derivative of (x^2)(sinx)(tanx)
Homework Equations
f(x+y)(d/dx)=(x)(y)d/dx+(y)(x)d/dx
That's for differentiating with 2 terms. With 3, I haven't done it yet.
The Attempt at a Solution
x^2(sinx)(sec^2x)+tanx(sinx)2x+x^2(tanx)(cosx)...
Homework Statement
Prove that the functions: (u+v)'(x0) and αu and u*v are derivable.
Homework Equations
in other words prove that :
(u+v)'(x_{0})=u'(x_{0})+v'(x_{0})
(\alpha u)'(x_{0})=\alpha u'(x_{0})
(u\cdot v)'(x_{0})=u'(x_{0})\cdot v(x_{0})+u(x_{0})\cdot v'(x_{0})
The...
Hello there!
I understand the product rule and how/why it works, and the proof makes sense,
but just out of curiosity, why would it be incorrect on a mathematical/application level to say that d/dx f(x)g(x) = f'(x)g'(x)
I know that it's wrong, and the product rule is the one that we're...
From my book:
Let
h = x + ay and g = x + by
By product rule we then have:
∂/∂x = ∂/∂h + ∂/∂g
Can someone explain to me how to arrive at this result? I don't even get what ∂/∂x even means, isn't it just a differential operator?
Hi, I am trying to simplify this;
x \frac{\partial}{\partial x} W(x,y) - \frac{\partial}{\partial x} (xW(x,y))
Am I correct in thinking I can do this with the product rule, as;
\frac{\partial}{\partial x} (xW(x,y)) = \left( \frac{\partial x}{\partial x}\right) W(x,y) + x...
Homework Statement
This is the problem and my attempt. I understand that I can take ln of both sides. However, I am trying to see if it is at all possible to arrive at the correct derivative without taking ln of both sides.
I am under the impression that an alternative to taking ln of...
I'm wondering what requirements must exist for a measure, m, to have the following properties:
m(Sum(P_i))= Sum(m(P_i)) the sum rule
m(Prod(P_i))=Prod(m(P_i)) the product rule
Where P_i are underlying sets or single propositions.
Thank you.
Homework Statement
Look up, figure out, or make an intelligent guess at the product rule for the scalar
product. That is, a rule of the form
d/dt [a(t).b(t)] =?+?
Verify your proposed rule on the functions
a(t) = ti + sin(t)j + e^(t)k and b(t) = cos(t)i - t^(2)j - e^(t)k:
Homework...
Homework Statement
f(x) = (x^3+6)^4 * (x^3+4)^6
Homework Equations
f'(x) = (x^3+6)^4*18x^2(x^3+4)^5 + (x^3+4)^6*12x^2(x^3+6)^3
The Attempt at a Solution
Obviously this expression can be simplified, but I haven't the slightest clue on how to do this.
I would really appreciate...
How would I find ∫[f(x)g(x)]d(x)? Similarly, how would I find ∫[f(x)/g(x)]dx?
Is there a similar rule to be applied here as in the product rule for differentiation?
I know there are other ways to obtain the product rule proff but my teacher got me started and I want to finish it this way and also have you guys explain WHY to do each step.
So far I have:
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
lim (F(x+h) - F(x))/h as h->0
lim (f(x+h)g(x+h) -...
Product rule generally seems straight forward but what if one comes across a scenario involving 3 functions instead of 2?
For example:
d/dx[x*e^(x^2)*f(x)]
f(x) is just some generic function
So there three functions of x are:
-x
- e^(x^2)
-f(x)
I am personally lost about how to...
Homework Statement
h(t) = √t (1 - t^2)
Homework Equations
the product rule is (first) x (derivative of the second) + (second) x (derivative of the first)
The Attempt at a Solution
i've been working at this for a while. the closest answer i came up with was this:
h(t)= t^1/2 (t^1/2 -...
i found a formula but I'm not sure is and maybe this is exist i don't know.
we know it F(x)= f(x).g(x) → F'(x)= f'(x).g(x)+g'(x).f(x)
and i want to calculate nth order.
n>1
fn(x).g(x)+ \sum^{n}_{k=1} binomial coefficient
\left(n k\right). f(n-k)(x).gk(x)+ gn.f(x)
is it true...
Hello,It is given that h(x) = f(x)g(x). It then tells me to write a formula for h'(2).
I know that h'(x) = f'(x)g(x) + f(x)g'(x), using the product rule.
So I assumed that h'(2) = f'(2)g(2) + f(2)g'(2)
Is this correct? Does the product rule simply allow me to do this? It seems to simple...
It's been years since I've done maths properly so I'm rusty with it. I'm helping out a colleague at work who is studying maths for an OU course.
Homework Statement
Part 1: Differentiate function.
f(x) = e^(0.5x+cos(x))
Part 2: Use answer from part 1 to show.
g(x) =...
Homework Statement
In general I havn't had problems using the differentiation rules until I came on this question, I'm probably doing something stupid any help is handy. Plugged it into an online differentiation solver and it comes up with (x^2-1)/(x^2) which I am getting nowhere near to in my...
Homework Statement
Use the Principle of Mathematical Induction and the Product Rule to prove the Power Rule when n is a positive integer.
Homework Equations
Dxxn = nxn-1
Dx(fg) = fDxg + Dxfg
The Attempt at a Solution
In summary,
Dxxn = nxn-1
Dxxk = kxk-1
Dxxk+1 = (k+1)x(k+1)-1
Dx(xkx) =...
Homework Statement
\frac{d}{dx}x^{2}y^{2} = ?
Homework Equations
Power rule: \frac{d}{dx}x^{n} = nx^{n-1}
Product rule: \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + g'(x)f(x)
The Attempt at a Solution
My TI-89 titanium gives me 2xy^{2}
Which contradicts the power rule.
Doing this by...
Hi I am 14 and attempting to learn calculus I have just proved product rule and am beginning examples of how it might work. Could anyone check, I will write my process.
y(x)=(12x^6)(7x^4+6)=
(12x^6)'(7x^4+6)+(12x^6)(7x^4+6)'=
(72x^5)(7x^4+6)+(12x^6)(28x^3)=
504x^9+432x^5+336x^9=
y(x)=840x^9+432x^5
If r and s are vectors that depend on time, prove that the product rule for differentiating products applies to r.s, that is that:
d/dt (r.s) = r. ds/dt + dr/dt .s
--
I'm not entirely sure how I'm supposed to go about proving this, can anyone point me in the right direction, please...
Homework Statement Use differentiation to verify that the following integrals are correct (where a is not = 0 is a constant and c is an arbitrary constant
(a) integrate xsinax dx= ( −x/a ) (cosax) +(1/a2) sinax+c
(b) integrate tanax dx=(−1/a) ln(cosax)+c
Homework Equations
Composite rule...
Differentiate and simplify:
y=(x+1)(2x-3)^{4}
I got:
8(x+1)(2x-3)^{3} + (2x-3)^{4}.
But the answers in the answer booklet say:
5(2x+1)(2x-3)^{3}
I put both answers in Wolfram Alpha and found they were both equal. So this is just a matter of simplifying/rearranging.
Could someone please...
Homework Statement
Using the product rule, differentiate the following function:
Homework Equations
y = etsintcost
The Attempt at a Solution
The three term product rule says:
d/dx (uvw) = u'vw + uv'w + uvw'
I find u = et, u' = et, v = sint, v' = cost, w = cost and w' = -sint...
Homework Statement
do what the title says
Homework Equations
The Attempt at a Solution
ok so I think it's
h(x)=f(x)g(x)
sum from k=0 to n of (n choose k) f^(n-k)(x) g^(k)(x)
right?
Homework Statement
Find the derivative of y= (2x2 + 1) x1/2
Homework Equations
Product RuleThe Attempt at a Solution
After differentiating, I eventually get stuck at:
\frac{4x3/2+2x2+1}{2\sqrt{x}}
The given solution is \frac{10x2+1}{2\sqrt{x}}
Out of curiosity, how does the product rule work in Lie groups? I ended up needing it because I approached a problem incorrectly and then saw that the product rule was unnecessary, but it seems to create a strange scenario. For example:
Consider a Lie group G and two smooth curves \gamma_1...
Wikipedia shows a proof of product rule using differentials by Leibniz. I am trying to correlate it to the definition of a differential and am having no success.
Differential Definition: http://eom.springer.de/D/d031810.htm
An example of implicit differentation in Stewart, 6th ed, p 883, is given as follows:
x^3 + y^3 + z^3 + 6xyz = 1
Differentiating to find dz/dx,
3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0
where the product rule was used to differentiate 6xyz with respect to x.
Why isn't the...
Homework Statement
Derive the following:
y = (cosxsin2x)-2
2. The attempt at a solution
Basically I saw this as a power rule with two products in the middle.
So y = -2 (cos2cos2x-sinxsin2x)-1
But the correct answer is completely different, it's:
4(3sin2x - 1) all over...
In trying to prove the limit product rule I've found all explanations
to hit on a point where I lose understanding.
1: If \lim_{x \to c} f(x) \ = \ L \ and \ \lim_{x \to c} g(x) \ = \ M \
We define the limit as;
\ \forall \ \epsilon \ >\ 0 \ \exists \ \delta > 0 \ : \...
[x_{\alpha}, p_{\alpha}]\psi(r)=[x_{\alpha}(-i\hbar \frac{\partial}{{ \partial x_\alpha}})-(-i\hbar\frac{\partial}{\partial x_{\alpha}})x_{\alpha}]\psi(r)
why the result is
i\hbar\psi(r) should not be 0?
and then the same situation
why in this case we get 0?
[x_{\beta}...
Homework Statement
Differentiate the following function with respect to x,
p(x) = (( x+5 )^2)*(( x+3 )^7)
Homework Equations
well the product rule is,
p(x)=(f)*(g)
p'(x)= (f')*(g)+(g')*(f)
and general differentiation is,
p'(x)=n(f)^(n-1)*n(g)^(n-1)
The Attempt at a Solution
well...
Homework Statement
r = r(t)
\theta = \theta(t)
x = r cos(\theta)
dx/dt =dr/dt cos(\theta) - r sin(\theta) d\theta/dtThe Attempt at a Solution
Where does the d\theta/dt come from at the end of the derivative? I know I'm using product rule here because r and theta are both functions of t...
My question is how do they transform equation A. into B.. I know they are using the product rule but don't know what is going on.
EQ A.)D(1/r^2)d/dr(r^2(dC/dr))-kC=0
now how do they get Eq B.
EQ B.) (d^2C/dr^2)+(2/r)(dC/dr)-(kC)/D=0
Hi, so I'm trying to solve Laplace's equation by separation of variables, and there's a basic step I'm not understanding with regards to the product rule.
Given
A product rule (i think) is taken to make the first term easier to deal with and we get
I'm just having trouble...
I'm trying to find the derivative of 0 = 3xcosƟ with respect to time.
I know I should use the product rule for x and cosƟ. But I don't know what I should do with the constant 3.
would it be like this?
0 = 3x(-sinƟ)(dƟ/dt) + 3(dx/dt)(cosƟ)
Homework Statement
Heya,
I can expand the term on the LHs and get the term on the RHS no problem :)
But,
I don't understand how the bottom line is arrived at; I think its basically just a backward engineering of the product rule, but I can't get it!
Any help would be useful
Homework Statement
i'm sorry i know I've been bombarding physics forums with questions but i need help :p
using reverse product rule \int uv' = uv - \int vu'
and say i have a*b
i noticed my teaher said that a=u and b=v not v' and he simly made that into a v' by deriving.
is there a point...
Homework Statement
f(x) = (2x-1)(3x-2)(5x+1) d/dx = ?
Note that I am letting (2x-1)(3x-2) = f and 5x+1 = g
Homework Equations
d/dx (fg) = f d/dx g + g d/dx f
The Attempt at a Solution
d/dx y = (2x-1)(3x-2)d/dx(5x+1)+(5x+1)d/dx((2x-1)(3x-2))
= (2x-1)(3x-2)(5)+...
Why are they similar?
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
d/dx (f(x)*g(x))=f(x)g'(x)+g(x)f'(x)
Somewhere on this very site there was mention of this, I believe, though I can't remember where. Maybe I'm delirious.
Hi,
So, I am reviewing Cal III, and I have come across something that I do not understand regarding implicit differentiation with partial derivatives:
x^3 + y^3 + z^3 + 6xyz = 1
implicit differentiation of z with respect to x:
3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0
*notive the...
Homework Statement
Question: the base b and height h change with a time t in such a way that b= (t+1)^{2} and h= t^{2}+1
Determine the rate of change of the area of the triangle when t=3
Homework Equations
A=\frac {1}{2}(b)(h)
The Attempt at a Solution
I am not sure what to do...
Wikipedia gives an extensive amount of vector identities:
http://en.wikipedia.org/wiki/Vector_calculus_identities
http://en.wikipedia.org/wiki/List_of_vector_identities
Does anyone know of a link where most of these are proved. I'm particularly interested in the product rules but would...