Derivative (examples)

Example 1

Consider "f"("x") = 5:

: $f\text{'}(x)=lim\_\{h\; ightarrow\; 0\}\; frac\{f(x+h)-f(x)\}\{h\}\; =\; lim\_\{h\; ightarrow\; 0\}\; frac\{f(x+h)-5\}\{h\}\; =\; lim\_\{h\; ightarrow\; 0\}\; frac\{(5-5)\}\{h\}\; =\; lim\_\{h\; ightarrow\; 0\}\; frac\{0\}\{h\}\; =\; lim\_\{h\; ightarrow\; 0\}\; 0\; =\; 0$

The derivative of a constant function is zero.

Example 2

Consider the graph of $f(x)=2x-3$. If the reader has an understanding of algebra and the Cartesian coordinate system, the reader should be able to independently determine that this line has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit, secant, and tangent) one can determine the slope at (4,5):

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The derivative and slope are equivalent.

Example 3

Via differentiation, one can find the slope of a curve. Consider $f(x)=x^2$:

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For any point "x", the slope of the function $f(x)=x^2$ is $f\text{'}(x)=2x$.

Example 4

Consider $f(x)\; =\; sqrt\{x\}$:

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Example 5

The same as the previous example, but now we search the derivative of the derivative.
Consider $f(x)\; =\; sqrt\{x\}$:

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