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The Derivative of a Power.
172. Problem. To find the derivative of any power of a variable.
Solution. Let the variable be a and the power an, and let 6 differ infinitely little from a; the derivative of an is then
b Now when b is equal to a, the value of this quotient is, by art. 51,
nan-1; and this must differ from the present value of this quotient, by an infinitely small quantity, which being neglected gives
for the derivative of an.
The derivative of any power of a variable is, therefore, found by multiplying by the exponent, and. diminishing the exponent by unity.
173. Corollary. The derivative of man when m is constant and a variable is n m an-1.
174. Problem. To find the derivative of any power of a function.
Solution. Let the variable be a, the function u, and the power un ; let b differ infinitely little from a, and let v be the corresponding value of u; if U is the derivative of u and U' that of un, we have
vn ” un
and U= 6
ba But, by art. 51,
The derivative of any power of a function is, therefore, found by multiplying by the exponent and by the derivative of the function, and diminishing the exponent by unity.
Find the erivatives of the following functions in which 2 is the variable.
Ans. 2 %. 2. 23.
Ans. 3 22. 3. 2 + ax + brp + &c.
Ans. n ra-1 +marm-1 +pbrp-' + &c. 4. A + Bx + C x2 + D 23 + E x4 + F25+ &c.
Ans. B+2 Cx+3 D x2 +4 E 23+5 Fz4+&c. 5. a + .
Ans. 1. 6. (a + x)
Ans. 2(a + x). 7. (a + x)3
Ans. 3 (a + x) 8. (a + x)".
Ans. n (a + x)*-1 9. (a + b x)2
Ans. 2 b(a+bx). 10. (a + b x)"
Ans. nb (a+bx)*-1.
176. Problem. To find the derivative of the product of two functions.
Solution. Let u and v be the functions, and U and V their derivatives ; then, since the derivative is the rate of change of the function to that of the variable, it is evident
The Derivative of a Product.
that when the variable is increased by the infinitesimal i, that the functions will become
u + U i and v + Vi. The product will therefore change from
(u + Ui) (v + Vi)=uv too Uitu Vi+ UV.3, and the increase of the product is
v Uitu Vi+ U Vis; the ratio of which to i is
v U+uV+U Vi, or, neglecting the infinitesimal U Vi, it is
v U +uV;
that is, the derivative of a product of two functions is equal to the sum of the two products obtained by multiplying each function by the derivative of the other function.
177. Corollary. The derivative of
(2-a)" v is, then,
n (2-a)n-10 + (x − a)* V, because the derivative of
(z – a)"
n (2 — a)-1
Solution of Numerical Equations.
178. Definition. A numerical equation is one all whose coefficients are given in numbers, so that it involves no literal expressions except those denoting the unknown quantities.
179. Problem. To solve a numerical equation.
Solution. Let the equation be reduced as in arts. 105 and 118, to the form
U = 0.
Find by trial a value of the unknown quantity x which nearly satisfies this equation, and let this value be a; substitute this value in the given equation, and let the corresponding value of u be m.
A correction e in the value of a is then to be found, which shall reduce the value of u from m to zero. Now, if U is the derivative of u, and if Mis the value of U which corresponds to
X = a, M is, by art. 165, the rate at which u changes in comparison with x, so that when
x = a te
u=m+ Me= 0, and therefore
By this means a value of x is found which is not
Rate of Approximation.
perfectly accurate, because M is not the rate at which u varies during the whole interval from
x = a to x=ate; but only while x differs infinitely little from a.
Calling, therefore, a' this approximate value of x, we have
M' which may
be used in the same way in which a was, in order to find a new approximate value a" of x; and if m' and M' denote the corresponding values of u and U, we shall have
M In the same way, may the approximation be continued to any degree of accuracy.
180. Problem. To determine the rate of approximation in the preceding solution.
Solution. This is a most important, practical point, and the determination of it will be found to facilitate the solution.
Now, it may be observed, that since e is the correction of a, its magnitude shows the degree of accuracy which belongs to a, and the accuracy of e is, obviously, the same with that of
a' = a + e. The comparative accuracy of the approximate value of a, and the succeeding approximate value a', is, then, the same with the magnitude of e compared with the error of e.
Now, in determining e, M was supposed to be the rate at which u changed throughout the whole interval in the change of a from a to a te. But if the rate of change of