Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUS◮Find from first principles the derivative with respect to x of f(x) =x 2 .Using the definition (2.1),f ′ f(x +∆x) − f(x)(x) = lim∆x→0 ∆x(x +∆x) 2 − x 2= lim∆x→0 ∆x2x∆x +(∆x) 2= lim∆x→0 ∆x= lim (2x +∆x).∆x→0As ∆x tends to zero, 2x +∆x tends towards 2x, hencef ′ (x) =2x. ◭Derivatives of other functions can be obtained in the same way. The derivativesof some simple functions are listed below (note that a is a constant):ddx (xn ) = nx n−1 ,dd(sin ax) = a cos ax,dxddx (tan ax) = a sec2 ax,ddx (eax ) = ae ax ,d(cos ax) = −a sin ax,dxddx (cot ax) = −a cosec2 ax,d(cos −1 x )−1= √dx a a2 − x ,2ddx (ln ax) = 1 x ,(sec ax) = a sec ax tan ax,dxd(cosec ax) = −a cosec ax cot ax,dxd(sin −1 x )1= √dx a a2 − x ,2d(tan −1 x )a=dx a a 2 + x 2 .Differentiation from first principles emphasises the definition of a derivative asthe gradient of a function. However, for most practical purposes, returning to thedefinition (2.1) is time consuming and does not aid our understanding. Instead, asmentioned above, we employ a number of techniques, which use the derivativeslisted above as ‘building blocks’, to evaluate the derivatives of more complicatedfunctions than hitherto encountered. Subsections 2.1.2–2.1.7 develop the methodsrequired.2.1.2 Differentiation of productsAs a first example of the differentiation of a more complicated function, weconsider finding the derivative of a function f(x) that can be written as theproduct of two other functions of x, namely f(x) =u(x)v(x). For example, iff(x) = x 3 sin x then we might take u(x) = x 3 and v(x) = sinx. Clearly the44