Differentiation Rules

In calculus, differentiation rules are formulas that allow us to find the derivative of a function easily. They are based on the definition of the derivative and simplify the process of differentiation.

Constant Rule

The derivative of a constant function is zero. If f(x) = c, where c is a constant, then f'(x) = 0.

Power Rule

The power rule is used when differentiating functions of the form f(x) = x^n, where n is any real number. The power rule states that f'(x) = n * x^(n-1).

Product Rule

The product rule is used when differentiating the product of two functions. If f(x) = g(x) * h(x), then the derivative of f is f'(x) = g'(x) * h(x) + g(x) * h'(x).

Quotient Rule

The quotient rule is used when differentiating the quotient of two functions. If f(x) = g(x) / h(x), then the derivative of f is f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]².

Chain Rule

The chain rule is used when differentiating the composition of two functions. If f(x) = g(h(x)), then the derivative of f is f'(x) = g'(h(x)) * h'(x).

Trigonometric Functions

The derivatives of the sine and cosine functions are: (sin(x))' = cos(x) and (cos(x))' = -sin(x). For the other trigonometric functions, their derivatives can be derived using the quotient rule.

Exponential and Logarithm Functions

The derivative of the natural exponential function is (e^x)' = e^x. The derivative of the natural logarithm function is (ln(x))' = 1/x. For other exponential and logarithm functions, their derivatives can be calculated using the chain rule.

These rules form the basis for differentiating a wide variety of functions. By combining these rules, it's possible to compute the derivatives of complex functions efficiently.