Derivaatiomenetelmät

Derivaatiomenetelmät refers to methods used to calculate the derivative of a function. The derivative of a function at a given point represents the instantaneous rate of change of the function at that point, or the slope of the tangent line to the function's graph at that point. Several fundamental methods exist for finding derivatives.

The power rule is a basic rule for differentiating polynomial functions. It states that the derivative of x^n is n*x^(n-1), where n is any real number. The constant multiple rule states that the derivative of c*f(x) is c times the derivative of f(x), where c is a constant. The sum and difference rules allow differentiation of sums and differences of functions by differentiating each term separately.

The product rule is used to find the derivative of a product of two functions. If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). The quotient rule is used for differentiating a quotient of two functions. If h(x) = f(x)/g(x), then h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2.

The chain rule is crucial for differentiating composite functions. If y = f(u) and u = g(x), then the derivative of y with respect to x is dy/dx = dy/du * du/dx. This means the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. These methods form the foundation for more complex differentiation techniques used in calculus.