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  2. Total derivative - Wikipedia

    en.wikipedia.org/wiki/Total_derivative

    The rate of change of f with respect to x is usually the partial derivative of f with respect to x; in this case, ∂ f ∂ x = y . {\displaystyle {\frac {\partial f}{\partial x}}=y.} However, if y depends on x , the partial derivative does not give the true rate of change of f as x changes because the partial derivative assumes that y is fixed.

  3. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function ⁠ ⁠, specifically the points (, ()) and (+, (+)). As h {\displaystyle h} is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to the graph of ...

  4. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    For instance, if f(x, y) = x 2 + y 2 − 1, then the circle is the set of all pairs (x, y) such that f(x, y) = 0. This set is called the zero set of f, and is not the same as the graph of f, which is a paraboloid. The implicit function theorem converts relations such as f(x, y) = 0 into functions.

  5. Automatic differentiation - Wikipedia

    en.wikipedia.org/wiki/Automatic_differentiation

    Currently, for its efficiency and accuracy in computing first and higher order derivatives, auto-differentiation is a celebrated technique with diverse applications in scientific computing and mathematics.

  6. Numerical differentiation - Wikipedia

    en.wikipedia.org/wiki/Numerical_differentiation

    A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h, h represents a small change in x, and it can be either positive or negative. The slope of this line is (+) ().

  7. Third derivative - Wikipedia

    en.wikipedia.org/wiki/Third_derivative

    In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function y = f ( x ) {\displaystyle y=f(x)} can be denoted by

  8. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.

  9. Rolle's theorem - Wikipedia

    en.wikipedia.org/wiki/Rolle's_theorem

    The red curve is the graph of function with 3 roots in the interval [−3, 2]. Thus its second derivative (graphed in green) also has a root in the same interval. The requirements concerning the n th derivative of f can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand ...