By Ivan Erdelyi
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Additional info for Operator theory and functional analysis
Sample text
Sometimes they are described explicitly where y equals some function of its independent variable(s), such as y = x sin x or implicitly where y, and its independent variable(s) are part of an equation, such as x 2 + y 2 = 10. A function may reference other functions, such as y = sin cos2 x or y = x sin x . There is no limit to the way functions can be combined, which makes it impossible to cover every eventuality. Nevertheless, in this chapter we explore some useful combinations that prepare us for any future surprises.
As an another example, let’s find dy/dx for x 2 − y 2 + 4x = 6y. 4 Differentiating Exponential and Logarithmic Functions 47 Differentiating, we have 2x − 2y dy dy +4=6 . dx dx Rearranging the terms, we have dy dy + 2y dx dx dy (6 + 2y) = dx 2x + 4 dy = . dx 6 + 2y 2x + 4 = 6 If, for example, we have to find the slope of x 2 − y 2 + 4x = 6y at the point (4, 3), then we simply substitute x = 4 and y = 3 in dy/dx to obtain the answer 1. Finally, let’s differentiate x n + y n = a n : x n + y n = an nx n−1 + ny n−1 dy =0 dx dy nx n−1 = − n−1 dx ny x n−1 dy = − n−1 .
Called e, where 1+ 1 n n e = lim 1+ 1 n nx ex = lim n→∞ . Raising e to the power x: n→∞ 48 4 Derivatives and Antiderivatives Fig. 8 Graphs of y = ex and y = e−x which, using the Binomial Theorem, is ex = 1 + x + x2 x3 x4 + + + ··· . 2! 3! 4! If we let y = ex dy x2 x3 x4 =1+x + + + + ··· dx 2! 3! 4! = ex . which is itself. 8 shows graphs of y = ex and y = e−x . Now let’s differentiate y = a x . We know from the rules of logarithms that log x n = n log x therefore, given y = ax then ln y = ln a x = x ln a therefore y = ex ln a which means that a x = ex ln a .