How to find the tangent line (linear approximation) of a function
In this HowDo, you will learn how to find the equation of the tangent line of a function $f(x)$ at a point $x = a$. Sometimes, we say that the tangent line is the linear approximation of the function $f(x)$ near $x=a$.
As an example, we shall find the derivative of $$f(x) = x^2 - 2$$ at the point $x = 2$. The equation of the tangent will be found to be $y = 4x-6$. So one may say that $y = 4x-6$ is the linear approximation of $f(x) = x^2 - 2$ near $x = 2$.
What is known
| a differentiable function | $f(x)$ |
| a point | $a$ |
What we need to calculate
| the value of $f$ at $x=a$ | $f(a)$ |
| the derivative of $f$ at $x=a$ | $f'(a)$ |
Step 1
Calculate $f(a)$
Remember that we use $f(x) = x^2 - 2$ and $a = 2$ for our example.
So we have $f(a) = a^2-2 = 2^2 - 2= 2$.
Step 2
Find the derivative $f'(x)$
We may use any method to find the derivative of $f(x)$.
In this example, we use standard formula to find $$f'(x) =(x^2 - 2)' =(x^2)' - 0 =2x.$$
Step 3
Find the value $f'(a)$
We know $f'(x) = 2x$ and $a = 2$, so $f'(a) = 2a = 4$.
Step 4
Write down the equation for the tangent
We know the slope of the tangent should be $f'(a)$, and it passes through the point $(a, f(a))$, so the equation is $$y-f(a) = f'(a)(x-a).$$
In our example, the equation becomes $$y - 2 = 4(x - 2).$$
Step 5
Rearrange to standard form
For our example, after rearrangement, we get the equation $$y = 4x-6.$$
So we know the tangent line of $f(x) = x^2 - 2$ at $x=2$ is given by $$y=4x-6.$$
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