Susan Nall

2021-12-12

Expand each function (using the appropiate technique/formula) Compute the derivative of the expanded function by applying the differentiation rules

$f\left(x\right)={(x+5)}^{2}$

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

kaluitagf

Beginner2021-12-13Added 38 answers

Step 1: To determine

Derivative of the given functions by expanding them:

1)$f\left(x\right)={(x+5)}^{2}$

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

Step 2: Formula used

1.$(x+y)}^{2}={x}^{2}+2xy+{y}^{2$

2.$(x-y)}^{2}={x}^{2}-xy+{y}^{2$

3.$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$

Step 3: Solution

Consider the given function:

$f\left(x\right)={(x+5)}^{2}$

Using formula, the expanded function is given by:

$f\left(x\right)={x}^{2}+2.x.5+{5}^{2}$

$\Rightarrow f\left(x\right)={x}^{2}+10x+25$

Differentiating the above function with respect to x, we get,

$\frac{df\left(x\right)}{dx}=\frac{d}{dx\mid ({x}^{2}+10x+25)}$

$\Rightarrow \frac{df\left(x\right)}{dx}=\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left(10x\right)+\frac{d}{dx}\left(25\right)$

$\Rightarrow \frac{df\left(x\right)}{dx}=2x+10\left(1\right)+0$

$\Rightarrow \frac{df\left(x\right)}{dx}=2x+10$

Hence, the derivative of the given function is$\frac{df}{dx}=2x+10$

Derivative of the given functions by expanding them:

1)

2)

Step 2: Formula used

1.

2.

3.

Step 3: Solution

Consider the given function:

Using formula, the expanded function is given by:

Differentiating the above function with respect to x, we get,

Hence, the derivative of the given function is

Chanell Sanborn

Beginner2021-12-14Added 41 answers

Step 4

Consider the given function:

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

Using formula, the expanded function is given by:

$f\left(x\right)=\left(4{x}^{2}\right)2-2.\left(4{x}^{2}\right)\left(3\right)+{3}^{2}$

$\Rightarrow f\left(x\right)=16{x}^{4}-24{x}^{2}+9$

Differentiating the above function with respect to x, we get,

$\frac{df\left(x\right)}{dx}=\frac{d}{dx}(16{x}^{4}-24{x}^{2}+9)$

$\Rightarrow \frac{df\left(x\right)}{dx}=\frac{d}{dx}\left(16{x}^{4}\right)+\frac{d}{dx}(-24{x}^{2})+\frac{d}{dx}\left(9\right)$

$\Rightarrow \frac{df\left(x\right)}{dx}=16\frac{d}{dx}\left({x}^{4}\right)-24\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left(9\right)$

$\Rightarrow \frac{df\left(x\right)}{dx}=16\left(4x3\right)-24\left(2x\right)+\left(0\right)$

$\Rightarrow \frac{df\left(x\right)}{dx}=64{x}^{3}-48x$

Hence, the derivative of the given function is$\frac{df}{dx}=64{x}^{3}-48x$

Consider the given function:

Using formula, the expanded function is given by:

Differentiating the above function with respect to x, we get,

Hence, the derivative of the given function is

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