Question 1131246
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We COULD do this:<br>
limit as h goes to 0 of (1/h)* {{{((x+h)^2-7)/((x+h)^2+3)-(x^2-7)/(x^2+3)}}}.<br>
But finding the derivative is going to be somewhat less work if we rewrite the function as<br>
{{{f(x) = 1-10/(x^2+3)}}}<br>
Then to find the derivative using increment steps, we do<br>
limit as h goes to 0 of (1/h)*{{{((1-10/((x+h)^2+3))-(1-10/(x^2+3)))}}}<br>
{{{((1-10/((x+h)^2+3))-(1-10/(x^2+3))) = 10/(x^2+3)-10/((x+h)^2+3) = 10(((x^2+2xh+h^2+3)-(x^2+3))/((x^2+3)((x+h)^2+3))) = 10((2xh+h^2)/((x^2+3)((x+h)^2+3)))}}}<br>
Then 1/h* {{{10((2xh+h^2)/((x^2+3)((x+h)^2+3)))}}} = {{{10((2x+h)/((x^2+3)((x+h)^2+3)))}}}<br>
And finally<br>
limit as h goes to 0 of {{{10((2x+h)/((x^2+3)((x+h)^2+3)))}}} = {{{20x/(x^2+3)^2}}}<br>