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.