Question

what is the derivative with respect to x of (x+1)³-x³
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Answer 1

SOLUTION

TO DETERMINE

The derivative with respect to x

$\displaystyle \sf{ {(x + 1)}^{3} - {x}^{3} }$

EVALUATION

Let y be given function

Then we have

$\displaystyle \sf{ y = {(x + 1)}^{3} - {x}^{3} }$

$\displaystyle \sf{ \implies y = {x}^{ 3} + 3 .{x}^{2}. 1 + 3.x. {1}^{2} + 1 - {x}^{3} }$

$\displaystyle \sf{ \implies y = {x}^{ 3} + 3 {x}^{2} + 3x + 1 - {x}^{3} }$

$\displaystyle \sf{ \implies y = 3 {x}^{2} + 3x + 1 }$

Differentiating both sides with respect to x we get

$\displaystyle \sf{ \implies \frac{dy}{dx} = \frac{d}{dx} ( 3 {x}^{2} + 3x + 1 ) }$

$\displaystyle \sf{ \implies \frac{dy}{dx} = \frac{d}{dx} ( 3 {x}^{2} ) + \frac{d}{dx} (3x )+ \frac{d}{dx} (1 ) }$

$\displaystyle \sf{ \implies \frac{dy}{dx} =3 \frac{d}{dx} ( {x}^{2} ) +3 \frac{d}{dx} (x )+ \frac{d}{dx} (1 ) }$

$\displaystyle \sf{ \implies \frac{dy}{dx} =3 .2x + 3.1 + 0 }$

$\displaystyle \sf{ \implies \frac{dy}{dx} =6x + 3 }$

$\displaystyle \sf{ \therefore \: \: \: \frac{d}{dx} \bigg[{(x + 1)}^{3} - {x}^{3} \bigg] =6x + 3 }$

FINAL ANSWER

$\displaystyle \sf{ \frac{d}{dx} \bigg[{(x + 1)}^{3} - {x}^{3} \bigg] =6x + 3 }$

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