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Section 3.1 : The Definition of the Derivative
3. Use the definition of the derivative to find the derivative of,
\[g\left( x \right) = {x^2}\]Show All Steps Hide All Steps
Start SolutionFirst 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.
Show Step 2Now 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,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{g'\left( x \right) = 2x}}\]