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Section 3.1 : The Definition of the Derivative

3. Use the definition of the derivative to find the derivative of,

\[g\left( x \right) = {x^2}\]

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First we need to plug the function into the definition of the derivative.

\[g'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{g\left( {x + h} \right) - g\left( x \right)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {x + h} \right)}^2} - {x^2}}}{h}\]

Make sure that you properly evaluate the first function evaluation. This is one of the more common errors that students make with these problems.

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Now all that we need to do is some quick algebra and we’ll be done.

\[g'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{{x^2} + 2xh + {h^2} - {x^2}}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{h\left( {2x + h} \right)}}{h} = \mathop {\lim }\limits_{h \to 0} \left( {2x + h} \right) = 2x\]

The derivative for this function is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{g'\left( x \right) = 2x}}\]