Proof of the Product Rule

Proof of the Product Rule in calculus is a fundamental concept that establishes the method for finding the derivative of the product of two functions. When dealing with functions that are multiplied together, the Product Rule provides a systematic approach to determine their rate of change. This rule is essential in calculus and plays a crucial role in solving problems related to optimization, rates of change, and curve analysis. By understanding the proof of the Product Rule, students and mathematicians gain a deeper insight into the principles that govern the differentiation of product functions, enhancing their ability to analyze complex mathematical relationships.

  • How do you derive a function composed of a division and a multiplication? Do you use the quotient rule or the product rule?
  • The sum of one number and the square of another number is #27#. What is the maximum product of these two numbers, and what two numbers give this product?
  • How do you prove that # int_0^1 \ x(1-x)^n \ dx = 1/((n+1)(n+2))# ?
  • How I do I prove the Product Rule for derivatives?
  • #| ( (x-1)^4, (x-1)^3, (x-1)^2, (x-1), 1 ), ( (x-2)^4, (x-2)^3, (x-2)^2, (x-2), 1 ), ( (x-3)^4, (x-3)^3, (x-3)^2, (x-3), 1 ), ( (x-4)^4, (x-4)^3, (x-4)^2, (x-4), 1 ), ( (x-5)^4, (x-5)^3, (x-5)^2, (x-5), 1 ) | = # ?
  • A curve has the equation #y=(x+2)sqrt(x-1)#. show that #dy/dx=(kx)/(sqrt(x-1))#, where k is a contant and state the value of k.?
  • #int_0^1 (x^3+x+1)/(x^2+2)^2 dx #?
  • X+y=50 which x and y are two positive numbers (a)for which numbers x is the product of the two numbers an increasing function of x (b)what is the maximum value of their product?
  • Proove that: #int _a^b f (x)dx= int_a^b f (a+b-x)dx# ?
  • Find the matrix X such that XA=B ..?
  • How do you calculate this limit without using l’Hospital’s rule: lim (x^2 + 5x - 4 / x+7 ) as x → 7 ?
  • If y=root1-x/1+x.then prove that (1-X)^2dy/dx+y=0?
  • Implicit domain how to prove #dy/dx# = #-(sqrt(1-y^2))/(sqrt(1-x^2)# ?
  • How can i derived ((1+x)^4/(x^2-2)^3)^2 in chain rule, quotient rule and product rule please? Thank you so much
  • What will be the differentiation of it?
  • How do you determine the coefficients #a# and #b# such that #p(x) =x^2+ax+b# satisfies #p(1) = -7# and #p'(1) = 4#?
  • The sum of two non negative numbers is 20.Find these numbers if (a)their product is as small as possible (b)their sum of their squares is as small as possible?
  • Kindly solve this? which option is correct?
  • Can somebody please help me work out the following? Differentiate the following products and quotients using algebraic simplification: (a) f(x) = x(1 - x + 3x^2) (b) f(x) = (4x - 2x^2 + 6) / 2x Thanks
  • How do you differentiate #f(x)=3x(2x+1)^3 # using the product rule?