Finding a Derivative of a Function

**jimmyp** · May 28th 2009, 09:55 PM

Hello again,

I'm again having some trouble using rules to solve down for a derivative of a function. Could someone please show me how I'd find the derivative of

$y=(\frac{4x^2}{3-x})^3$

That would be great. Thanks for reading!

**Chris L T521** · May 28th 2009, 10:23 PM

Originally Posted by jimmyp

Hello again,

I'm again having some trouble using rules to solve down for a derivative of a function. Could someone please show me how I'd find the derivative of

$y=(\frac{4x^2}{3-x})^3$

That would be great. Thanks for reading!

This is a chain rule problem.

Let's approach it this way. Let $u=\frac{4x^2}{3-x}$ . It follows that $y=u^3$ . Now by chain rule, $\frac{\,dy}{\,dx}=\frac{\,dy}{\,du}\cdot\frac{\,du }{\,dx}$ .

Since we let $u=\frac{4x^2}{3-x}$ , it follows by quotient rule that $\frac{\,du}{\,dx}=\frac{(3-x)(8x)-4x^2(-1)}{(3-x)^2}=\frac{24x-4x^2}{(3-x)^2}$ .

Therefore,

$\frac{\,dy}{\,dx}=\underbrace{3u^2}_{\frac{\,dy}{\ ,du}}\cdot \underbrace{\frac{24x-4x^2}{(3-x)^2}}_{\frac{\,du}{\,dx}}=3\left(\frac{4x^2}{3-x}\right)^2\cdot\frac{24x-4x^2}{(3-x)^2}$

This can be simplified as $\frac{48x^4\cdot4x\left(6-x\right)}{\left(3-x\right)^4}=\frac{192x^5\left(6-x\right)}{\left(3-x\right)^4}$

Does this make sense?

**TheEmptySet** · May 28th 2009, 10:24 PM

Originally Posted by jimmyp

Hello again,

I'm again having some trouble using rules to solve down for a derivative of a function. Could someone please show me how I'd find the derivative of

$y=(\frac{4x^2}{3-x})^3$

That would be great. Thanks for reading!

First note that this is function composition

$f(x)=x^3,g(x)=\frac{4x^2}{3-x}$

$y=f(g(x))=\left(\frac{4x^2}{3-x}\right)^3$

before we use the chain rule lets calculate the needed derviatves

$f'(x)=3x^2$

$g'(x)=\frac{(3-x)(8x)-(4x^2)(-1)}{(3-x)^2}=\frac{24x-8x^2+4x^2}{(3-x)^2}=\frac{24x-4x^2}{(3-x)^2}=\frac{4x(6-x)}{(3-x)^2}$

Now the chain rule tells us that

$y'=f'(g(x))\cdot g'(x)=3\left(\frac{4x^2}{(3-x)} \right)^2\left( \frac{4x(6-x)}{(3-x)^2}\right)=\frac{192x^5(6-x)}{(3-x)^3}$

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