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 \(x\) .