8 APPLIED MECHANICS 
likely to be required in ordinary engineering problems. Those in the 
first and second lines occur most frequently. 
y=C dy _ hohe [uan=cx 
da 
d: . =e 
yaa Y= naxn 1 ear" fude= am 
(except when n= — 1) 
y=a logx So auiae u=ag-1= [ude log « 
1 
ya qY — abe uae [uae = fo 
y=a sin ba at EP cos bx u=a cos be [ude = sin bx 
dx b 
d : : 
y=a cos bx ah= — ab sin ba u=a sin ba [ude =~ $ 00s bx 
dy _ ap 2h 2 a 
y=a tan bx ister ee u=a sec bx Jude = ¢ tan bx 
y=a cot bx Bs —ab cosec? bz u=a cosec? bx [uae = -¢ cot bx 
The differential coefficient of a constant is zero. 
The differential coefficient of the sum of a number of functions is the 
sum of the differential coefficients of the functions. 
Thus, if y=w+v+w, where w, v, and w are functions of x, then 
dic die de dee’ 
The differential coefficient of the product of a number of functions is 
found by multiplying the differential coefficient of each factor by all the 
other factors and adding the products thus formed. Thus, if y=wvw, 
dy _,,, du dv, dw 
dz dz da da 
The differential coefficient of the quotient of two functions is found 
as follows: From the product of the denominator and the differential 
coefficient of the numerator subtract the product of the numerator and 
the differential coefficient of the denominator, “8 divide the result by 
where u, v, and w are functions of x, then 
the square of the denominator. Thus, if y= 7, where ~ and v are 
y_ oft 
functions of 2, ae dx dz 
Hi - eee * 
Function of a function.—If y is a function of uw, and w is a function 
dy _dy du 
of z, then 7 =— +a For example, let y= /at+tba+cx. Put 
