Hence
d y d x = x ( a − 2 x ) 2 b ( a − x ) x + ( a − x ) x b = x ( 3 a − 4 x ) 2 b ( a − x ) x {\displaystyle {\frac {dy}{dx}}={\frac {x(a-2x)}{2b{\sqrt {(a-x)x}}}}+{\frac {\sqrt {(a-x)x}}{b}}={\frac {x(3a-4x)}{2b{\sqrt {(a-x)x}}}}} .
Now
(We shall need these two last differential coefficients later on. See Ex. X. No. 11)
Exercises VI. (See page 257 for Answers.)
Differentiate the following:
(1) y = x 2 + 1 {\displaystyle y={\sqrt {x^{2}+1}}} .
(2) y = x 2 + a 2 {\displaystyle y={\sqrt {x^{2}+a^{2}}}} .
(3) y = 1 a + x {\displaystyle y={\dfrac {1}{\sqrt {a+x}}}} .
(4) y = a a − x 2 {\displaystyle y={\dfrac {a}{\sqrt {a-x^{2}}}}} .
(5) y = x 2 − a 2 x 2 {\displaystyle y={\dfrac {\sqrt {x^{2}-a^{2}}}{x^{2}}}} .
(6) y = x 4 + a 3 x 3 + a 2 {\displaystyle y={\dfrac {\sqrt[{3}]{x^{4}+a}}{\sqrt[{2}]{x^{3}+a}}}} .
(7) y = a 2 + x 2 ( a + x ) 2 {\displaystyle y={\dfrac {a^{2}+x^{2}}{(a+x)^{2}}}} .