Differentiation: Gradient Function

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

Differentiation of Special Functions

  • 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.

Applying the Gradient Function

  • 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.

Practice Problems

  • 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.