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Differentiate of the Following from First Principle: E3x - Mathematics

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Differentiate  of the following from first principle:

e3x

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Solution

\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( e^{3x} \right) = \lim_{h \to 0} \frac{e^{3(x + h)} - e^{3x}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{3x} e^{3h} - e^{3x}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{3x} \left( e^{3h} - 1 \right)}{3h}\]
\[ = 3 e^{3x} \lim_{h \to 0} \frac{e^{3h} - 1}{3h}\]
\[ = 3 e^{3x} \left( 1 \right)\]
\[ = 3 e^{3x}\]

 

Concept: The Concept of Derivative - Algebra of Derivative of Functions
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 2.02 | Page 25

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