Product Rule for Derivatives
Read this topicExplanations 18
- I often like limiting cases when I derive equations. What we want to know is a way to take the derivative of the product of two functions ddx(f(x)g(x))\frac{d}{dx} (f(x) g(x))dx…
By G. Edward Marti
- The derivative of a product of two functions is equal to the sum of the first function times the derivative of the second function plus the derivative of the first function time…
By Ylaine Gerardin
- (This explanation uses little-o notation.) Suppose functions f(x)f(x)f(x), g(x)g(x)g(x) are differentiable at aaa. What can we say about (f(x)g(x))′∣x=a(f(x)g(x))' |_{x=a}(f(x)g…
By Victor Wang, Po-Shen Loh, Expii Team, Alex Federspiel
- The derivative of product of functions f(x)f(x)f(x) and g(x)g(x)g(x) is equal to the product of f(x)f(x)f(x) and the derivative of g(x)g(x)g(x) plus the product of the derivativ…
By Harrison Grodin, Alex Federspiel
- Although the Product Rule is most commonly seen for the expansion of a two-factor product, if there are more than two factors, the Product Rule can be extended for these more co…
By Joshua Siktar, Ahaan Rungta, Alex Federspiel
- This is an intuitive explanation of the product rule using a physical example... Imagine you're tiling a floor with little square tiles that have side length sss. You have a til…
By Elena Koslover, Alex Federspiel
- Suppose we have a function we don't know how to take the derivative of, but the function is the product of two functions we do know how to take the derivative of, like this: h(x…
By Matt Superdock, Ahaan Rungta
- PROOF OF PRODUCT RULE: ddx[f(x)g(x)]=limf(x+a)g(x+a)−f(x)g(x)a=limf(x+a)g(x+a)−f(x)g(x+a)+f(x)g(x+a)−f(x)g(x)a=limg(x+a)(f(x+a)−f(x))a+limf(x)(g(x+a)−g(x))a=f′(x)g(x)+f(x)g′…
By Kevin Ren, Alex Federspiel
- (This will possibly need some knowledge of combinatorics, induction, and binomial coefficients. This is very optional, and for the very interested reader. :) ) In this post we w…
By Yang Liu, Alex Federspiel
- One might think that the derivative of a product is the product of the derivatives in question. However, this is wrong. An example of this is: Let f(x)=xf(x)=xf(x)=x and f(g)=x2…
By Michael U
- We know how to find the derivative of functions f(x)f(x)f(x) and g(x)g(x)g(x). The derivatives are f′(x)f'(x)f′(x) and g′(x)g'(x)g′(x) respectively. But what about the derivativ…
By Kevin Yang, Moor Xu
- Here are some extra practice problems, have fun solving them!
By Freya Edholm
- (Videos) Using Chain, Product, and Quotient Rules to Find Derivatives by PatrickJMT Here's a video by patrickJMT showing you how to use the chain, product, and quotient rules to…
By Alex Federspiel
- (Video) The Product Rule for Derivatives by PatrickJMT Here's a video by patrickJMT showing you how to use the product rule to find a derivative of a function. The product rule …
By Alex Federspiel
- (Video) Proof of the Product Rule by PatrickJMT Curious as to how we came up with the product rule? Luckily, patrickJMT has a video just for you! Here he uses the definition of …
By Alex Federspiel
- (Videos) Using Chain, Product, and Quotient Rule to Find Derivatives by PatrickJMT Here's a video by patrickJMT showing you how to find the derivative of functions where you nee…
By Alex Federspiel
- (Video) Complicated Derivative Examples by PatrickJMT Here's a video by patrickJMT going over some derivative problems that have a mix of different rules. The first problem he l…
By Alex Federspiel
- (Video) Solving Every Derivative Ever by PatrickJMT Okay, maybe it's not EVERY derivative ever but here's a video by patrickJMT going over different derivatives for a whopping 3…
By Alex Federspiel
Problems 16
- What is the derivative of xsinxx \sin xxsinx?
- What's the derivative of (x+3)sinx(x+3) \sin x(x+3)sinx?
- If f(x)=x∗x,f(x) = x * x,f(x)=x∗x, what does the product rule say about the derivative of f(x)f(x)f(x)?
- What is the derivative of xe−xxe^{-x}xe−x
- Let fff be a three-times differentiable real function such that f(x2)⋅f′′(x)=f′(x)⋅f′(x2)f(x^2)\cdot f''(x) = f'(x)\cdot f'(x^2)f(x2)⋅f′′(x)=f′(x)⋅f′(x2) for all real xxx. Given…
- Using the product rule, find the derivative of f(x)=x2f(x)=x^2f(x)=x2?
- What is the derivative of x4cos(x)x^4\cos(x)x4cos(x)?
- Find the derivative of f(x)f(x)f(x). f(x)=(1x3−2x2)(4x4+3x)f(x)=\bigg(\frac1{x^3}-\frac2{x^2}\bigg)(4x^4+3x)f(x)=(x31−x22)(4x4+3x)
- Find the derivative of f(x)f(x)f(x). f(x)=(x)(ln(x)+3)f(x)=(\sqrt x)(\ln(x)+3)f(x)=(x)(ln(x)+3)
- Find the derivative of f(x)f(x)f(x). f(x)=(x+5)(x−5)f(x)=(x+5)(x-5)f(x)=(x+5)(x−5)
- Find the derivative of f(x)f(x)f(x). f(x)=(x+1)(x2+2)(x3+3)f(x)=(x+1)(x^2+2)(x^3+3)f(x)=(x+1)(x2+2)(x3+3)
- Find the intervals on which the function f(x)=x2x+2f(x)=x^2\sqrt{x+2}f(x)=x2x+2 is increasing and decreasing.
- Find the intervals on which the function f(x)=x2lnx2f(x)=x^2\ln x^2f(x)=x2lnx2 is increasing and decreasing.
- Find the intervals on which the function f(x)=xex(x2ex)f(x)=xe^x(x^2e^x)f(x)=xex(x2ex) is increasing and decreasing.
- Mark all functions fff below such that (f⋅g)′=f⋅g′(f\cdot g)' = f\cdot g'(f⋅g)′=f⋅g′ for all differentiable functions ggg.
- What is the derivative of xsinxx\sin xxsinx?