Yet Another Calculus Text, 2007a

30 CHAPTER 1. DERIVATIVES
Exercise 1.7.5. Find the derivative of y =13x 5 .
Example 1.7.7. Combining the power rule with our results for constant multiples
and differences, we have
d
dx (3x2 − 5x) =6x − 5.
Exercise 1.7.6. Find the derivative of f(x) =5x 4 − 3x 2 .
Exercise 1.7.7. Find the derivative of y =3x 7 − 3x +1.
1.7.4 Polynomials
As the previous examples illustrate, we may put together the above results to
easily differentiate any polynomial function. That is, if n ≥ 1anda n , a n−1 ,
..., a 0 are any real constants, then
d
dx (a nx n + a n−1 x n−1 + ···+ a 2 x 2 + a 1 x + a 0 )
= na n x n−1 +(n − 1)a n−1 x n−2 + ···+2a 2 x + a 1 . (1.7.26)
Example 1.7.8. If p(x) =4x 7 − 13x 3 − x 2 + 21, then
p ′ (x) =28x 6 − 39x 2 − 2x.
Exercise 1.7.8. Find the derivative of f(x) =3x 5 − 6x 4 − 5x 2 + 13.
1.7.5 Quotients
If u is a differentiable function of x, u(x) ≠0,anddx is an infinitesimal, then
( 1 1
d =
u)
u(x + dx) − 1
u(x)
1
=
u(x)+du − 1
u(x)
u − (u + du)
=
u(u + du)
−du
=
u(u + du) . (1.7.27)
Hence, since u + du ≃ u, ifdx ≠0,
( ) du
d 1
= − dx
dx u u(u + du) ≃−1 du
u 2 dx . (1.7.28)