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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 above expansion holds because the derivative of e x with respect to x is also e x, and e 0 equals 1. This leaves the terms (x − 0) n in the numerator and n! in the denominator of each term in the infinite sum.
If f is a function, then its derivative evaluated at x is written ′ (). It first appeared in print in 1749. [3] Higher derivatives are indicated using additional prime marks, as in ″ for the second derivative and ‴ for the third derivative. The use of repeated prime marks eventually becomes unwieldy.
From any of these definitions it can be shown that e −x is the reciprocal of e x. For example, from the differential equation definition, e x e −x = 1 when x = 0 and its derivative using the product rule is e x e −x − e x e −x = 0 for all x, so e x e −x = 1 for all x.
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 ) {\textstyle \arctan(y,x)} .
One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function f(x) at the point x 0 is a linear polynomial a + b(x − x 0), and it may be possible to get a better approximation by considering a quadratic polynomial a + b(x − x 0) + c(x − x 0) 2.
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.}
In calculus, the power rule is used to differentiate functions of the form () =, whenever is a real number.Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule.