Changes

(Proving this is a worth while exercise).

(Proving this is a worth while exercise).

For example, if <math>y = 3 x^2</math>, the derivative with respect to <math>x</math> is  

For example, if <math>y = 3 x^2+2x</math>, the derivative with respect to <math>x</math> is  

:<math>\frac{dy}{dx} = 6 x</math>

:<math>\frac{dy}{dx} = 6 x+2</math>

Thus the derivative is a measurement of how a function changes when the values of its inputs vary. Derivatives are helpful in determining the [[maximum|maxima]] and [[minimum|minima]] of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called ''critical'' points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing. Also, differentials have numerous applications in physics.

Thus the derivative is a measurement of how a function changes when the values of its inputs vary. Derivatives are helpful in determining the [[maximum|maxima]] and [[minimum|minima]] of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called ''critical'' points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing. Also, differentials have numerous applications in physics.