By M. Batty

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**Extra resources for Essential Engineering Mathematics**

**Example text**

This is useful because we can more easily find powers and limits of a matrix representing T . With respect to an eigenbasis the matrix of a transformation is diagonal, meaning all entries not on the leading diagonal are 0. For example, λ1 0 A= 0 λ2 is diagonal, and An is just λn1 0 0 λn2 There is a process for calculating eigenvalues and eigenvectors which relies on (a) calculating determinants (b) row reduction. Eigenvalues λ of an n×n matrix A, and corresponding eigenvectors v satisfy Av = λv, which can be rewritten as (A − λIn )v = 0.

Product Rule: f g is differentiable and (f g) = f g + gf . 4. Quotient Rule: f /g is differentiable (where defined) and gf − f g . 2 1. e use a limit argument as with x2 above. ) 2. Use the above rules to differentiate x1999 + x2 + 1 . 2 The Chain Rule The chain rule tells us how to differentiate the composition of two functions. The Chain Rule: If f and g are functions from R to R, f is differentiable at x ∈ R and g is differentiable at f (x) then g o f is differentiable at x and (g ◦ f ) (x) = g (f (x))f (x) This theorem is also true for functions only defined on subsets of R, provided that we can compose them.

Xp−1 (by the product rule) = (p − 1 + 1)xp−1 = pxp−1 . d (1) = 0 everywhere (this can’t be incorporated into the If p = 0, dx power rule if x = 0). 56 Calculus of One Variable Essential Engineering Mathematics Suppose that p ∈ Z and p < 0. Let q = −p. Then q > 0 and we use the quotient rule. 1 x2q −q = xq+1 = pxp−1 . = Thus the power rule holds for all p ∈ Z. To extend it to rational n we can use the chain rule. m x m dx m−1 by the chain rule. Hence n d xm dx n xn−1 . x = m n n −1 xm = m = pxp−1 .