Question

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!

Answer 1

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 )}}}}\\\\`