Question

Find the derivative of x³-27

Answer 1

let y=x^3 - 27

then,

dy/dx = d (x^3 - 27)/dx

by properties of differtiation,

dy/dx = 3x^2

Answer 2

Hey!
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→ We can find the derivative of x³ - 27 by using 2 ways :


First. → using the first principle ,


Let y = x³ - 27 ---------------( 1 )

Then , ∆y + y = ( x + ∆x )³ - 27 ------------( 2 )


Subtracting ( 2 ) - ( 1 )

→ ∆y = x³ + ∆x³ + 3x•∆x² + 3x²•∆x + ∆x³ + 27 - x³ - 27


✔cancel Out the Like terms ,


→ ∆y = ∆x³ + 3x²∆x + 3x∆x²


✔ Diving the equation by ∆x and put Lim. ∆x → 0 :


=> ∆y/∆x = Lim. ∆x→ 0 ∆x³ + 3x²∆x + 3x∆x² / ∆x

{ Take ∆x common and cancel it on numerator and denominator }


=> dy/dx = ∆x² + 3x² + 3x∆x


=> dy/dx = 3x² { as ∆x = 0 }

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Second. → is the direct way. Use the formula dy/dx for xⁿ = n x^n-1 and derivative of constant is 0.


=> 3 ( x² ) - 0


=> 3x²


You can opt for any method as per choice !!


°•° The derivative of x³-27 = 3x²
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