(i) cos – (sin x ) (ii) tan − (iii) − Solution (i) Let f ( x ) = cos – (sin x ). Observe that this function is defined for all real numbers. We may rewrite this function as f ( x ) = cos – (sin x ) = cos π = π − Thus f ′ ( x ) = – . (ii) Let f ( x ) = tan – .
Observe that this function is defined for all real numbers, where cos x ≠ – ; i.e., at all odd multiplies of π . We may rewrite this function as f ( x ) = tan − sin tan 2cos tan tan − = Observe that we could cancel cos in both numerator and denominator as it is not equal to zero. Thus f ′ ( x ) = . (iii) Let f ( x ) = sin – .
To find the domain of this function we need to find all x such that −≤ . Since the quantity in the middle is always positive, we need to find all x such that , i.e., all x such that x + ≤ + x . We may rewrite this as ≤ x + x which is true for all x . Hence the function is defined at every real number.
By putting x = tan θ , this function may be rewritten as f ( x ) = − = sin − + ( 2tan tan = sin – [sin θ ] = θ = tan – ( x ) Thus f ′ ( x ) = ( ) ( )log2 x ⋅ log2 Example Find f ′ ( x ) if f ( x ) = (sin x ) sin x for all < x < π . Solution The function y = (sin x ) sin x