# Expand each function (using the appropiate technique/formula) Compute the derivative

Expand each function (using the appropiate technique/formula) Compute the derivative of the expanded function by applying the differentiation rules
$f\left(x\right)={\left(x+5\right)}^{2}$
$f\left(x\right)={\left(4{x}^{2}-3\right)}^{2}$
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kaluitagf
Step 1: To determine
Derivative of the given functions by expanding them:
1) $f\left(x\right)={\left(x+5\right)}^{2}$
2) $f\left(x\right)={\left(4{x}^{2}-3\right)}^{2}$
Step 2: Formula used
1. ${\left(x+y\right)}^{2}={x}^{2}+2xy+{y}^{2}$
2. ${\left(x-y\right)}^{2}={x}^{2}-xy+{y}^{2}$
3. $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$
Step 3: Solution
Consider the given function:
$f\left(x\right)={\left(x+5\right)}^{2}$
Using formula, the expanded function is given by:
$f\left(x\right)={x}^{2}+2.x.5+{5}^{2}$
$⇒f\left(x\right)={x}^{2}+10x+25$
Differentiating the above function with respect to x, we get,
$\frac{df\left(x\right)}{dx}=\frac{d}{dx\mid \left({x}^{2}+10x+25\right)}$
$⇒\frac{df\left(x\right)}{dx}=\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left(10x\right)+\frac{d}{dx}\left(25\right)$
$⇒\frac{df\left(x\right)}{dx}=2x+10\left(1\right)+0$
$⇒\frac{df\left(x\right)}{dx}=2x+10$
Hence, the derivative of the given function is $\frac{df}{dx}=2x+10$
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Step 4
Consider the given function:
$f\left(x\right)={\left(4{x}^{2}-3\right)}^{2}$
Using formula, the expanded function is given by:
$f\left(x\right)=\left(4{x}^{2}\right)2-2.\left(4{x}^{2}\right)\left(3\right)+{3}^{2}$
$⇒f\left(x\right)=16{x}^{4}-24{x}^{2}+9$
Differentiating the above function with respect to x, we get,
$\frac{df\left(x\right)}{dx}=\frac{d}{dx}\left(16{x}^{4}-24{x}^{2}+9\right)$
$⇒\frac{df\left(x\right)}{dx}=\frac{d}{dx}\left(16{x}^{4}\right)+\frac{d}{dx}\left(-24{x}^{2}\right)+\frac{d}{dx}\left(9\right)$
$⇒\frac{df\left(x\right)}{dx}=16\frac{d}{dx}\left({x}^{4}\right)-24\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left(9\right)$
$⇒\frac{df\left(x\right)}{dx}=16\left(4x3\right)-24\left(2x\right)+\left(0\right)$
$⇒\frac{df\left(x\right)}{dx}=64{x}^{3}-48x$
Hence, the derivative of the given function is $\frac{df}{dx}=64{x}^{3}-48x$