Proof of Quotient Rule
The quotient rule stands as a fundamental principle in calculus, offering a systematic method for differentiating functions that are expressed as the quotient of two other functions. Derived from the basic principles of differentiation, the quotient rule provides a clear and concise approach to finding the derivative of such complex functions. Its proof, rooted in the principles of limits and algebraic manipulation, elucidates the rationale behind its formulation and underscores its applicability in solving a wide array of mathematical problems. Understanding the proof of the quotient rule is essential for mastering calculus and applying its principles effectively in various mathematical and scientific contexts.
Questions
- Show that? #(P(N', n))/(P(N,n)) ~~ ((N')/N)^n# #(N', N, n) in ZZ^+# #(N', N )">>" n# #P(x,y) = (x!)/((x-y)!)# Stirling Approx allowed
- How I do I prove the Quotient Rule for derivatives?
- How do you differentiate #f(x)=(2x^2-4x-1)/(x-5)# using the quotient rule?
- How do you solve this problem? Please explain step by step.
- If #f= x^2 y^2+2xy^2#, then show that #(partial ^2f)/(partial x partial y) = (partial ^2f)/(partial y partial x)#. obtain the value of #f_(xy)(1,1)# ?
- How do you differentiate #f(x)=(cos^3x)/sinx# using the quotient rule?
- #lim_(x -> 0) 1/x (2/(2+x)^3-1/4) = # ?
- If f(x)=(4x+5)/(5x+6), how do you find f'(x)?
- How I do I prove the Chain Rule for derivatives?
- Does anybody know how to solve the limit of #(x^x-x^{x^2})/(1-x)^2# when #x->1# ?
- What is the maximum value of #Q(p,q,r)=2pq+2pr+2qr# subject to #p+q+r=1#?
- How do you differentiate #f(x)=(2x)/(sqrt(5x^2 -2x + 1))# using the quotient rule?
- How do you show that #sum_(k=0)^(n-1) z^k = (1 - z^n)/(1-z)# ?
- How do you differentiate #f(x)=x^2 / (1-x)^(1/2)# using the quotient rule?
- If #h(x)=p(x) * q(x) * r(x)# what is #h'#?
- If #u= y/x+ z/x + x/y#, show that #x(partial u)/(partial x)+y(partial u)/(partial y)+z(partial u)/(partial z)=0# ?
- Prove that f is invertible and find (f^-1)'((1)/(2) ?
- Prove, using the definition of the derivative, that# (a f(x))'=a f'(x)# (where #a# is constant with respect to #x#)?
- How do you differentiate #f(x)=(x^2-3x+2)/(x-3)# using the quotient rule?
- How do you differentiate #f(x)=(x^3+x)/(4x+1)# using the quotient rule?