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Derivative of a Function \(f (x)\) at a Number \(a\)
$$f'(a)=\lim\limits_{h\to0}\frac{f(a+h)-f(a)}{h}\text{, or}f'(a)=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a},$$
provided that the limit exists. (Note that the former formula is used more often than the latter.)
Example: Using the definition of the derivative at a number, find the slope of tangent line at x = -2 to the curve y = -6x2. What is the equation of the tangent line at this point?
Solution: By definition of the derivative, the slope of the tangent line at x = -2 is
$$\begin{aligned}
f^{\prime}(-2)& =\operatorname*{lim}_{h\to0}{\frac{f(-2+h)-f(-2)}{h}} \\
&=\lim_{h\to0}\frac{-6(-2+h)^2-(-6(-2)^2)}h \\
&=\lim_{h\to0}\frac{-24+24h-6h^2+24}h \\
&=\lim_{h\to0}\frac{h(24-6h)}h=\lim_{h\to0}(24-6h)=24
\end{aligned}$$
When \( x = -2, y(-2) = -6(-2)^2 = -24 \), that is, the point of tangency is \((-2, -24)\).
Using the point-slope formula, $$\begin{aligned}y+24&=24(x+2)\\y&=24x+24\end{aligned}.$$
Derivative Function \(f '(x)\) of a Function \(f (x)\)
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h\text{, for any \(x\) for which this limit exists.}$$
Example: Using the definition of the derivative, given the function \(f(x)=3-\frac{1}{x^2},\text{find}f'(x).\)
Solution: By the definition of the derivative,
$$\begin{aligned}
f^{\prime}(x)& =\operatorname*{lim}_{h\to0}{\frac{f(x+h)-f(x)}{h}} \\
&=\lim_{h\to0}\frac{\left[3-\frac1{\left(x+h\right)^{2}}\right]-\left[3-\frac1{x^{2}}\right]}h=\lim_{h\to0}\frac{-\frac1{\left(x+h\right)^{2}}+\frac1{x^{2}}}h \\
&=\lim_{h\to0}\frac{\frac{-x^2+(x+h)^2}{x^2(x+h)^2}}h=\lim_{h\to0}\bigg(\frac{-x^2+x^2+2xh+h^2}{x^2(x+h)^2}\bigg)\bigg(\frac1h\bigg) \\
&=\lim_{h\to0}\frac{h(2x+h)}{hx^2(x+h)^2}=\lim_{h\to0}\frac{2x+h}{x^2(x+h)^2}=\frac{2x}{x^2(x)^2}=\frac2{x^3}
\end{aligned}$$