Differentiate \frac{r}{\sqrt{r^2+5}} ?

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I'm unsure how they went from the first step to the second step? How did the (r^2+5)^(1/2) on the denominator from the first step turn into an exponent of 1 on the second step? Also, how do you get the -r^2 in the numerator in the second step?

2 Answers
Jun 12, 2018

5/(sqrt((r^2+5)^3)

Explanation:

"differentiate using the "color(blue)"product rule"

"given "y=g(h(x))" then"

dy/dx=g(x)h'(x)+h(x)g'(x)larrcolor(blue)"product rule"

"here "y=r/(sqrt((r^2+5)^(1/2)))=r(r^2+5)^(-1/2)

g(r)=r rArrg'(r)=1

h(r)=(r^2+5)^(-1/2)

h'(r)=-1/2(r^2+5)^(-3/2)xxd/(dr)(r^2+5)

color(white)(h'(x))=-1/2(r^2+5)^(-3/2)xx2r=-r(r^2+5)^(-3/2)

(dy)/(dr)=-r^2(r^2+5)^(-3/2)+(r^2+5)^(-1/2)

color(white)((dy)/(dr))=(r^2+5)^(-3/2)[cancel(-r^2)cancel(+r^2)+5]

color(white)((dy)/(dr))=5/(r^2+5)^(3/2)

Jun 12, 2018

By expanding by sqrt(r^2+5)/sqrt(r^2+5)

Explanation:

Your first question can be answered by a simple algebra rule. We know that :

(a^m)^n = a^(m*n)

so therefore ((r^2+5)^(1/2))^2 = (r^2+5)^(2*1/2) = (r^2+5)

In the next step, we expanded the fraction by sqrt(r^2+5)/sqrt(r^2+5).

Therefore in the denominator we got (r^2+5)^(3/2) and the second part of the numerator became :

-r^2*(r^2+5)^(-1/2)*(r^2+5)^(1/2) = -r^2.