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Find the equation of the line tangent to the graph of $f\left(x\right)=1\text{/}x$ at $x=2.$
We can use [link] , but as we have seen, the results are the same if we use [link] .
We now know that the slope of the tangent line is $-\frac{1}{4}.$ To find the equation of the tangent line, we also need a point on the line. We know that $f\left(2\right)=\frac{1}{2}.$ Since the tangent line passes through the point $(2,\frac{1}{2})$ we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation $y=-\frac{1}{4}x+1.$ The graphs of $f\left(x\right)=\frac{1}{x}$ and $y=-\frac{1}{4}x+1$ are shown in [link] .
Find the slope of the line tangent to the graph of $f\left(x\right)=\sqrt{x}$ at $x=4.$
$\frac{1}{4}$
The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the derivative . The process of finding a derivative is called differentiation .
Let $f(x)$ be a function defined in an open interval containing $a.$ The derivative of the function $f(x)$ at $a,$ denoted by ${f}^{\prime}\left(a\right),$ is defined by
provided this limit exists.
Alternatively, we may also define the derivative of $f(x)$ at $a$ as
For $f\left(x\right)={x}^{2},$ use a table to estimate ${f}^{\prime}(3)$ using [link] .
Create a table using values of $x$ just below $3$ and just above $3.$
$x$ | $\frac{{x}^{2}-9}{x-3}$ |
---|---|
$2.9$ | $5.9$ |
$2.99$ | $5.99$ |
$2.999$ | $5.999$ |
$3.001$ | $6.001$ |
$3.01$ | $6.01$ |
$3.1$ | $6.1$ |
After examining the table, we see that a good estimate is ${f}^{\prime}\left(3\right)=6.$
For $f\left(x\right)={x}^{2},$ use a table to estimate ${f}^{\prime}(3)$ using [link] .
$6$
For $f\left(x\right)=3{x}^{2}-4x+1,$ find ${f}^{\prime}(2)$ by using [link] .
Substitute the given function and value directly into the equation.
For $f\left(x\right)=3{x}^{2}-4x+1,$ find ${f}^{\prime}(2)$ by using [link] .
Using this equation, we can substitute two values of the function into the equation, and we should get the same value as in [link] .
For $f\left(x\right)={x}^{2}+3x+2,$ find ${f}^{\prime}\left(1\right).$
${f}^{\prime}\left(1\right)=5$
Now that we can evaluate a derivative, we can use it in velocity applications. Recall that if $s\left(t\right)$ is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval $\left[a,t\right]$ if $t>a$ or $\left[t,a\right]$ if $t<a$ is given by the difference quotient
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