Derivative question

Splint

Hi,

I'm having a little trouble with finding the derivative of this one. The fraction with the square root is giving me the trouble, I'm not sure if I should be using the quotient rule or the chain rule or the chain rule within the quotient rule.
Anyone able to clarify this for me?

Thanks
David

$$\displaystyle y=60-\frac{x}{5}+ \frac{\sqrt{100^2+x^2}}{3}$$

drumist

Since the denominator of the fraction is a constant, it's easiest to merely factor out the constant. What I mean is:

$$\displaystyle \frac{d}{dx} \left[ \frac{\sqrt{100^2+x^2}}{3} \right]= \frac{1}{3} \cdot \frac{d}{dx} \left[ \sqrt{100^2+x^2} \right]$$

At this point you should use the chain rule where $$\displaystyle u=100^2+x^2$$ to give

$$\displaystyle \frac{1}{3} \cdot \frac{d}{du} \left[ u^{1/2} \right] \cdot \frac{d}{dx} \left[100^2+x^2 \right]$$