More Differentiation

More Differentiation

Understanding the Basic Principles of Differentiation

  • Differentiation measures how a function changes as its input changes.
  • Differentiating a function gives the derivative, which can be used to find the gradient of a tangent to the curve at any given point.
  • The basic rules of differentiation are crucial: the derivative of a constant is zero, the derivative of x^n is n*x^(n-1)

The Power Rule

  • To differentiate functions of the form f(x) = x^n, we use the power rule: d/dx[x^n] = nx^(n-1).
  • Note: The power rule only applies for constant exponents n.

Differentiating Polynomials

  • Larger polynomials can offer more of a challenge, but can be solved by applying the power rule term by term.
  • If there is a constant term, remember it will differentiate to zero.

Stationary Points

  • Stationary points are points where the derivative is zero.
  • Three types of stationary points are: maximum, minimum and points of inflection.

Increasing and Decreasing Functions

  • A function is increasing when its derivative is positive and decreasing when its derivative is negative.

Convex and Concave Curves

  • Curves are concave up where their second derivative is positive and concave down where the second derivative is negative.

Chain Rule

  • Chain Rule is used to differentiate composite functions.
  • For a composite function, y = f(g(x)), the derivative is dy/dx = f’(g(x)) * g’(x).

Differentiating e^x, ln x and a^x

  • The derivative of e^x is still e^x.
  • The derivative of ln x is 1/x.
  • For any constant a, the derivative of a^x = a^x ln a.

Differentiating sin, cos and tan

  • The derivative of sin(x) is cos(x), and cos(x) differentiates to -sin(x).
  • The derivative of tan(x) is sec^2(x) or 1/cos^2(x)

Product and Quotient Rules

  • The Product Rule is used to differentiate the product of two functions.
  • The Quotient Rule is used to differentiate the quotient of two functions.

More Differentiation Techniques

  • Implicit differentiation is used when it is difficult to express y explicitly as a function of x.
  • Differentiation can also be used to solve some types of differential equations.