Q. 3 G4.5( 4 Votes )

# Differentiate the following from first principles

Answer :

We need to find derivative of f(x) = x^{2} e^{x}

Derivative of a function f(x) is given by –

f’(x) = {where h is a very small positive number}

∴ derivative of f(x) = x^{2} e^{x} is given as –

f’(x) =

⇒ f’(x) =

⇒ f’(x) =

Using algebra of limits, we have –

⇒ f’(x) =

⇒

As 2 of the terms will not take indeterminate form if we put value of h = 0, so obtained their limiting value as follows –

∴ f’(x) = 0×e^{x + 0} + 2x e^{x + 0} +

Use the formula:

⇒ f’(x) = 2x e^{x} + x^{2} e^{x}

⇒ f’(x) = 2x e^{x} + x^{2} e^{x}

Hence,

Derivative of f(x) = x^{2} e^{x} = 2x e^{x} + x^{2} e^{x}

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