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In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
The mean value theorem gives a relationship between values of the derivative and values of the original function. If f ( x ) is a real-valued function and a and b are numbers with a < b , then the mean value theorem says that under mild hypotheses, the slope between the two points ( a , f ( a )) and ( b , f ( b )) is equal to the slope of the ...
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as ...
In mathematics, differential refers to several related notions [1] derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. [2] The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}
for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of the second lines.
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}