find df/dx

**dapore** · December 24th 2009, 11:16 PM

if

find $\frac{\mathrm{d} f}{\mathrm{d} x}$

**mr fantastic** · December 25th 2009, 03:05 AM

Originally Posted by dapore

if

find $\frac{\mathrm{d} f}{\mathrm{d} x}$

You need to use the chain rule multiple times, and set the work out carefully.

Chain rule 1: Let $u = x + \sqrt{x + \sqrt{x}}$ .

Then $y = \sqrt{u}$ and $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .

Chain rule 2: To get $\frac{du}{dx}$ , you will need to let $v = x + \sqrt{x}$ .

Then $u = x + \sqrt{v}$ and $\frac{du}{dx} = 1 + \frac{d \sqrt{v}}{dx} = 1 + \frac{d \sqrt{v}}{dv} \cdot \frac{dv}{dx}$ .

Now put it all together.

**tom@ballooncalculus** · December 25th 2009, 03:16 AM

Just in case a picture helps...

... because...

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

__________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

**dapore** · December 30th 2009, 01:24 AM

Thank you my friends

**wonderboy1953** · December 31st 2009, 11:10 AM

Square both sides of the equation and then take the derivative of the resultant equation (which is actually easier to do).

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