The basic rule for differentiation is the limit definition of a derivative, which states that the derivative of a function f at a point x is the limit of the difference quotient as the change in x approaches zero. However, this definition is often impractical for direct computation. Therefore, several differentiation rules have been developed to simplify the process.
The most basic differentiation rule is the power rule, which states that if f(x) = x^n, where n is a constant, then the derivative f'(x) is given by f'(x) = nx^(n-1). This rule is particularly useful for polynomials and other functions that can be expressed as powers of x.
Another important rule is the constant multiple rule, which states that the derivative of a constant times a function is the constant times the derivative of the function. In other words, if c is a constant and f(x) is a differentiable function, then (cf(x))' = c * f'(x).
The product rule and the quotient rule are essential for differentiating products and quotients of functions, respectively. The product rule states that if f(x) and g(x) are differentiable functions, then (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). The quotient rule states that if f(x) and g(x) are differentiable functions, then (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.
The chain rule is a powerful tool for differentiating composite functions. It states that if f(x) is a differentiable function and g(x) is a differentiable function, then the derivative of the composite function f(g(x)) is given by (f(g(x)))' = f'(g(x)) * g'(x). This rule is particularly useful for functions that are built up from simpler functions.
Finally, the derivative of certain standard functions, such as trigonometric, exponential, and logarithmic functions, are known and can be used as building blocks for differentiating more complex functions. These include the derivatives of sine, cosine, tangent, cotangent, secant, and cosecant, as well as the derivatives of the natural exponential function and the natural logarithm.