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








{\displaystyle (u\cdot v)'=u'\cdot v+u\cdot v'}
or in Leibniz's notation as











{\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.

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  1. mcastillo356

    Calculus Confusion over Calculus Book example footnote

    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...
  2. karush

    Differentiate ##f(x)=x\cos{x}+2\tan{x}: D/dx ##\tiny{2.4.2}##

    ##\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
  3. karush

    MHB -apc.2.2.03 trig product rule

    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 $1\cdot \sin (x)+\cos (x)x$...
  4. O

    I Proving the product rule using probability

    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...
  5. WhiteWolf98

    Triple Product Rule Equivalency

    ##p=\frac {RT} v;~p=p(T,v)~...1## ##v=\frac {RT} p;~v=v(T,p)~...2## ##T=\frac {pv} R;~T=T(p,v)~...3## ##Considering~eq.~1:## ##p=\frac {RT} v \Rightarrow (\frac {\partial p} {\partial v})_T=-\frac {RT} {v^2}## ##Considering~eq.~2:## ##v=\frac {RT} p \Rightarrow (\frac {\partial v}...
  6. FreeThinking

    A Why does MTW keep calling the "product rule" the "chain rule"?

    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...
  7. opus

    Is this a typo? Using Chain Rule and Product Rule

    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...
  8. navneet9431

    Problem in applying the Chain Rule

    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...
  9. P

    How many number of IP addresses are available?

    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...
  10. J

    B Product rule OR Partial differentiation

    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...
  11. P

    LaTeX Generalized Product Rule D^(n-m) (x^2 -1)^n (LaTeX inside)

    Homework Statement Show: ##D^{(n-m)} (x^2-1)^n = \frac{(n-m)!}{(n+m)!} (x^2-1)^m D^{(n+m)} (x^2-1)^n## Hint: ##D^{(n-m)} (x^2-1)^n = D^{(n-m)} [(x-1)^n (x+1)^n]## Homework Equations [/B] Leibniz Rule for Differentiation: $$D^k (uv) = \sum_{j=0}^k \binom{k}{j} D^j (u) D^{(k-j)} (v)$$ The...
  12. O

    Derivative of two polynomials, one of them being squared

    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.
  13. P

    I Product rule for exterior covariant derivative

    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...
  14. V

    I Product rule in Newton notation?

    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?
  15. P

    Why not use the product rule to expand Newton's 2nd law?

    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...
  16. Destroxia

    What is the significance of this curl product rule?

    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...
  17. O

    Mastering the Product Rule: Solving Problems with Ease | Homework Statement

    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.
  18. O

    Derivative of Dot Product via Product Rule, commutative?

    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...
  19. A

    Proof of product rule for gradients

    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
  20. DiracPool

    Product rule of a trig function

    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...
  21. A

    Understanding the Application of Chain and Product Rules in Calculus

    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.
  22. UMath1

    Product Rule Proof: Reasons Why it Won't Work

    Why wouldn't this work?
  23. J

    Proof of the logarithm product rule

    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...
  24. B

    Product rule for vector derivative

    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...
  25. BiGyElLoWhAt

    Excellent video series raises good question:

    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...
  26. Muthumanimaran

    Product rule in differentiation

    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.
  27. C

    Can Calculus Problems be Solved by Factoring?

    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]
  28. karush

    MHB Using the Product Rule to Solve $\d{}{x}{3}^{x}\ln\left({3}\right)$

    $\d{}{x}{3}^{x}\ln\left({3}\right)=$ I tried the product rule but didn't get the answer😖
  29. L

    MHB Simplifying the Product Rule for Derivatives

    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...
  30. M

    Checking simple differential product rule

    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...
  31. S

    Differentiating using a product rule

    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...
  32. S

    Proof of distributive and product rule

    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...
  33. Greg Bernhardt

    Lie derivatives: ##L_Xf=[X,f]##The product rule Definition/Summary - What is it?

    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))...
  34. G

    Proof of Quotient Rule using Product Rule

    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?
  35. anthonyk2013

    How Can the Derivative of y=3x²ln(x) Be Simplified Further?

    y=3x2Inx u=3x2 v=Inx du/dx=6x dv/dx=I/x y=u*v dy/dx=(u)(dv/dx)+(v)(du/dx) dy/dx=(3x2)(I/x)+(Inx)(6x) Can I simplify this further
  36. MarkFL

    MHB Hp1357's question at Yahoo Answers regarding an extended product rule

    Here is the question: I have posted a link there to this thread so that the OP may view my work.
  37. AntSC

    Quotient & Product Rule: Same funtion, different answers?

    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...
  38. T

    Product Rule Shortcut for Complicated Derivatives

    Find y' y=(x2+1)7(x9+2)5(x3+1)3(x8+7)3 Is there a shortcut to doing this problem? Or do I have to actually use the product rule more than 3 times?
  39. D

    MHB Partial Derivatives of Matrix/Vector Function: An Easier Way?

    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...
  40. D

    Using the product rule for the partial derivative of a vector function

    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...
  41. B

    Which rules to use (Product rule?)

    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...
  42. A

    Product rule in probability and more

    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...
  43. L

    What is the proof for the Product Rule in Calculus?

    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...
  44. T

    Hyperbolic function and the product rule.

    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...
  45. J

    Product rule of derivative of expectation values

    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...
  46. A

    The product rule and Bayes rule

    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...
  47. Petrus

    MHB Understanding the Chain Rule and Product Rule in Multivariable Calculus

    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...
  48. S

    Integrating a Fraction using the product rule

    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...
  49. D

    Derivative Help: differentiate using product rule

    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)
  50. H

    Product Rule Homework: Find (fg)'(2)

    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 :(