Differentiation: Gradient Function

Understanding Differentiation and Gradient Functions

Differentiation is one key concept in calculus. It describes the rate of change of one quantity with respect to another. It helps to understand how a quantity changes as it varies.
The derivative of a function is often referred to as the gradient function because it gives the slope or gradient of the tangent to the function at any point.
The basic rule to differentiate a power of x is known as the Power Rule: if y = x^n, then dy/dx = n.x^n-1. The ‘n’ moves in front of x and the power is reduced by 1.
When a constant is attached to a function, we can differentiate it normally and then multiply by that constant. If y = 5x^n, then dy/dx = 5n.x^n-1

Differentiating a constant: The derivative of a constant is zero because a constant doesn’t change: if y = c where c is a constant, then dy/dx = 0.
Differentiating a linear function: For a linear function y = mx, the derivative is just the coefficient of x, m: dy/dx = m.
Differentiating the sum or difference of functions: The derivative of a sum or difference of functions is the sum or difference of their derivatives.

The derivative of a function at a particular point can be used to find the equation of the tangent to the function at that point. The gradient of the tangent is given by the derivative, and the y-intercept can be found by substituting the x-value of the point into the equation of the tangent.
Similarly, the derivative can also be used to find the equation of the normal, or the line perpendicular to the tangent, at a point.
The second derivative, or the derivative of the derivative, can be used to determine concavity and points of inflexion of a function. If the second derivative is positive at a point, the function is concave up there; if it’s negative, the function is concave down there; and if it changes sign, the function has a point of inflexion there.

Always test your understanding by attempting various differentiation problems. It is advisable to work on problems in sections, beginning with the basics and moving on gradually to more complex functions.
Familiarise with the usage of calculus in real-life applications by trying to solve word problems involving calculus. Remember to interpret your answers in the context of each problem.