Differentiate the Following Functions from First Principles X2ex ? - Mathematics

Question

Differentiate the following functions from first principles x²e^x ?

Sum

Solution

$Let f (x) = x^{2} e^{x}$

$\Rightarrow f (x + h) = {(x + h)}^{2} e^{(x + h)}$

$= lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$= lim_{h \to 0} \frac{{(x + h)}^{2} e^{(x + h)} - x^{2} e^{x}}{h}$

$= lim_{h \to 0} (\frac{x^{2} e^{x + h} - x^{2} e^{x}}{h} + \frac{2 x h e^{(x + h)}}{h} + \frac{h^{2} e^{(x + h)}}{h})$

$= lim_{h \to 0} (\frac{x^{2} e^{x} (e^{(x + h) - x} - 1)}{h} + 2 x e^{(x + h)} + h e^{(x + h)})$

$= lim_{h \to 0} (x^{2} e^{x} \frac{(e^{h} - 1)}{h} + 2 x e^{(x + h)} + h e (x + h))$

$= x^{2} e^{x} + 2 x e^{x} + 0 x e^{x} [∵ lim_{x \to 0} \frac{e^{x} - 1}{x} = 1]$

$∴ \frac{d}{d x} (x^{2} e^{x}) = e^{x} (x^{2} + 2 x)$

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Simple Problems on Applications of Derivatives

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Chapter 11: Differentiation - Exercise 11.01 [Page 17]

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Chapter 11 Differentiation
Exercise 11.01 | Q 8 | Page 17

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Differentiate the Following Functions from First Principles X2ex ? - Mathematics

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