x = sin y . Differentiating both sides w.r.t. x , we get = cos y dy which implies that cos(sin Observe that this is defined only for cos y ≠ , i.e., sin – x ≠ , π π , i.e., x ≠ – , , i.e., x ∈ (– , ). To make this result a bit more attractive, we carry out the following manipulation.
Recall that for x ∈ (– , ), sin (sin – x ) = x and hence cos y = – (sin y ) = – (sin (sin – x )) = – x Also, since y ∈ , π π , cos y is positive and hence cos y = Thus, for x ∈ (– , ), f ( x ) sin – x cos - x tan - x Domain off (- , ) (- , ) R f ( x ) EXERCISE . Find dy dx in the following: . x + y = sin x . x + y = sin y .
ax + by = cos y . xy + y = tan x + y . x + xy + y = . x + x y + xy + y = .
sin y + cos xy = κ . sin x + cos y = . y = sin – . y = tan – , .
, sec − . Exponential and Logarithmic Functions Till now we have learnt some aspects of different classes of functions like polynomial functions, rational functions and trigonometric functions. In this section, we shall learn about a new class of (related) functions called exponential functions and logarithmic functions. It needs to be emphasized that many statements made in this section are motivational and precise proofs of these are well beyond the