Application of the derivative

• One of the most important uses of calculus is determining stationary (minimum, maximum, inflection) points, sometimes called turning points.
• If we consider \(y=x^{2}\), we know that we can find the gradient of the tangent at different values of \(x\) by substituting into the derivative of the function, which is \(\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=2x\).

Notice at the stationary point, the gradient of the tangent is zero (when \(x=0\), \( \frac{\mathrm{d}y}{\mathrm{d}x} = 0\)), the progression of the tangents show a minimum turning point.