100. Differentiate "log tan-1x with regard to 2x with regard to sin-1 1 + 22° sin x 3 106. If S, the sum of a G. P. to n terms of which r is the 114. Given the identity (2cos 20-1) (2cos 220-1)... (2cos 2"0-1)= prove that r=n 2'sin 20 r1 2cos20 - 1 2"+1sin 2"+10 2sin 20 115. Given sinno sino sin(2a +) sin(4a+p)...sin {2(n-1)a + } =· 2n-1 coto+cot (2a +4)+ cot(4a + ) + ... + cot{2(n - 1)a+} and that cosec ++cosec2(2a + ) + cosec2(4a+)+ ... 116. Given =ncotno, + cosec3 {2(n-1)a + p} = n'cosecno. 222 )( 02 322 CHAPTER IV. SUCCESSIVE DIFFERENTIATION. 88. Repeated Operations. d dx The operation denoted by is defined in Art. 39 with out any reference to the form of the function operated upon, the only assumption made being that the function is a function of the same independent variable as that referred to in the operative symbol, viz. x. It is moreover clear that the result of the operation is also a function of x, and as such is itself capable of being operated upon by the same symbol. That is to say, if y be a function is also a function of x, and therefore we can have dy dx of x, d (dy as a true mathematical quantity. And further, it dx dx d will be thus seen that the operation may be performed dx upon any given function of x any number of times. 89. Notation. The expression d (dy dx dx is generally abbreviated into |