Introduction to derivatives

Summary

• $f$ is a given function and $x_0$ is a point in the domain of $f$ . $\left( x_0,f\left( x_0 \right) \right)$ is then a point on the graph of $f$ .
• A straight line that goes through the point $\left( x_0,f\left( x_0 \right) \right)$ that “ touches ” the graph at this point is called a tangent of $f$ at $x=x_0$ . (see below for strict definition of tangent).
• The slope of the tangent of $f$ at $x=x_0$ is called the derivative of $f$ at $x=x_0$ , denoted by $f'\left( x_0 \right)$ .

The derivative of $f$ is a function denoted by $f'$ . $f'\left( x \right)$ is the slope of the tangent of $f$ at an arbitrary point $x$ .