Question

Find the derivatives: f(x)=\frac{x^{2}+2x+5}{x^{2}+1}

Derivatives
ANSWERED
asked 2021-06-13
Find the derivatives:
\(f(x)=\frac{x^{2}+2x+5}{x^{2}+1}\)

Answers (1)

2021-06-14

Step 1
Given function
\(f(x)=\frac{x^{2}+2x+5}{x^{2}+1}\)
Using quotien rule of the derivative
\(f y=\frac{u}{v}\ then\ y'=\frac{vu'-uv'}{v^{2}}\)
Step 2
The derivative of f with respect to x.
\(f'(x)=\frac{(x^{2}+1)\frac{d}{dx}(x^{2}+2x+5)-(x^{2}+2x+5)\frac{d}{dx}(x^{2}+1)}{(x^{2}+1)^{2}}\)
\(f'(x)=\frac{(x^{2}+1)(2x+2)-(x^{2}+2x+5)(2x)}{(x^{2}+1)^{2}}\)
\(f'(x)=\frac{2x^{3}+2x+2x^{2}+2-2x^{3}-4x^{2}-10x}{(x^{2}+1)^{2}}\)
\(f'(x)=\frac{-2x^{2}-8x+2}{(x^{2}+1)^{2}}\)

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