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Explore how the derivative functions as a variable changes, differentiation operators, differentiability of functions, and higher-order derivatives in calculus.
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2.2 The derivative as a function In Section 2.1 we considered the derivative at a fixed number a. Now let number a vary. If we replace number a by a variable x, then the derivative can be interpreted as a function of x : Alternative notations for the derivative: D and d / dxare called differentiation operators. dy / dx should not be regarded as a ratio.
Differentiable functions A function f is differentiable at a if f ′(a) exists. It is differentiable on an open interval(a,b) [ or (a,) or (- , a) or (- , ) ] if it is differentiable at every number in the interval. Theorem: If f is differentiable at a, then f is continuous at a. Note: The converse is false: there are functions that are continuous but not differentiable. Example: f(x) = | x |
To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp discontinuity vertical tangent
is the first derivative of y with respect to x. is the second derivative. is the third derivative. is the fourth derivative. Higher Order Derivatives: (y double prime) We will learn later what these higher order derivatives are used for.
