215k views
4 votes
Find dy/dx if y =x^3+5x+2/x²-1

How would I go about finding this? I would appreciate if you could be as detailed as possible!

Find dy/dx if y =x^3+5x+2/x²-1 How would I go about finding this? I would appreciate-example-1
User Cania
by
9.2k points

1 Answer

8 votes

Differentiate using the Quotient Rule


\qquad
\pink{\twoheadrightarrow \sf (d)/(dx) \bigg[(f(x))/(g(x)) \bigg]= ( g(x)\:(d)/(dx)\bigg[f(x)\bigg] -f(x)(d)/(dx)\:\bigg[g(x)\bigg])/(g(x)^2)}\\

According to the given question, we have –

  • f(x) = x^3+5x+2
  • g(x) = x^2-1

Let's solve it!


\qquad
\green{\twoheadrightarrow \bf (d)/(dx)\bigg[ (x^3+5x+2 )/(x^2-1)\bigg]} \\


\qquad
\twoheadrightarrow \sf ((x^2-1) (d)/(dx)(x^3+5x+2) - ( x^3+5x+2) (d)/(dx)(x^2-1))/((x^2-1)^2 )\\


\qquad
\twoheadrightarrow \sf ((x^2-1)(3x^2+5) - ( x^3+5x+2) 2x)/((x^2-1)^2 )\\


\qquad
\pink{\sf \because (d)/(dx) x^n = nx^(n-1) }\\


\qquad
\twoheadrightarrow \sf (3x^4+5x^2-3x^2-5-(2x^4+10x^2+4x))/((x^2-1)^2 )\\


\qquad
\twoheadrightarrow \sf (3x^4+5x^2-3x^2-5-2x^4-10x^2-4x)/((x^2-1)^2 )\\


\qquad
\green{\twoheadrightarrow \bf (x^4-8x^2-4x-5)/((x^2-1)^2 )}\\


\qquad
\pink{\therefore \bf{\green{\underline{\underline{(d)/(dx) (x^3+5x+2 )/(x^2-1)} = (x^4-8x^2-4x-5)/((x^2-1)^2 )}}}}\\\\

User Guillaume Perrot
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories