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A Function; its Variable, and Rate of Change.
164. Definition. When quantities are so connected that their values are dependent upon each other, each is said to be a function of the others; which are called variables when they are supposed to be changeable in their values, and constants when they are supposed to be unchangeable.
y is a function of the a, b, and x; but if x is variable while a and b are constant, it is more usual to regard y as simply a function of x.
165. Definition. In the case of a change in the value of a function, arising from an infinitely small change in the value of one of its variables, the relative rate of change of the function and the variable, that is, the ratio of the change in the value of the function to that in the value of the variable, is called the derivative of the function.
The derivative of the derivative of a function is called the second derivative of the function; the derivative of the second derivative is called the third derivative; and so on.
166. Corollary. The derivative of a constant is
167. Corollary. The derivative of the variable, regarded as a function of itself, is unity; and the second derivative is zero.
The Derivative of the sum of any Functions.
The derivative of the sum of two
functions is the sum of their derivatives.
Proof. Let the two functions be u and v, and let their values, arising from an infinitesimal change i in the value of their variable, be u' and v'; the increase of their sum will be
which is obviously the sum of their derivatives.
169. Corollary. By reversing the sign of v, it may be shown, in the same way, that the derivative of the difference of two functions is the difference of their derivatives.
170. Corollary. The derivative of the algebraic sum of several functions connected by the signs + and is the algebraic sum of their derivatives.
171. Corollary. If, in this sum, any function is repeated any number of times, its derivative must be repeated the same number of times; in other words, if a function is multiplied by a constant its derivative must be multiplied by the same constant.
Thus, if the derivatives of u, v, and w are respectively U, V, and W, and if a, b, c, and e are constant, the derivative of
The Derivative of a Power.
power of a variable.
To find the derivative of any
Solution. Let the variable be a and the power a", and let 6 differ infinitely little from a; the derivative of an is
Now when bis equal to a, the value of this quotient is, by art. 51,
n an -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 a".
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 ̧ ̧
To find the derivative of any
power of a function.
Solution. Let the variable be a, the function u, and the power u"; 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 u”, we have
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 derivatives of the following functions in which x is the variable.
Ans. 2 x.
Ans. 3 x2.
3. xn + a xm + b x2 + &c.
Ans. n xn−1+m a xTM-1+ p b xr-1+ &c. -1 +maxm-1+p
4. A+B+C x2 + D x3 + E x2 + F x5+&c. Ans. B+2Cx+3D x2+4 Ex3 +5 Fx1+&c.
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+Ui and v + Vi.
The product will therefore change from
(u+Ui) (+Vi)=moto VitaVi+UV,
and the increase of the product is
v U i + u Vi+ U V 22;
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