(log x ) log x , x > . cos ( a cos x + b sin x ), for some constant a and b . . (sin x – cos x ) (sin x – cos x ) , π π .
x x + x a + a x + a a , for some fixed a > and x > . x x − + , for x > . Find dy , if y = ( – cos t ), x = ( t – sin t ), π π < < . Find dy dx , if y = sin – x + sin – , < x < .
If , for , – < x < , prove that ) = − . If ( x – a ) + ( y – b ) = c , for some c > , prove that is a constant independent of a and b . . If cos y = x cos ( a + y ), with cos a ≠ ± , prove that cos ( .
. If x = a (cos t + t sin t ) and y = a (sin t – t cos t ), find . . If f ( x ) = | x | , show that f ″ ( x ) exists for all real x and find it.
. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines. . Does there exist a function which is continuous everywhere but not differentiable at exactly two points?
Justify your answer. . If g x h x l m n b , prove that f g x h x l m n b ′ ′ ′ . If y = , – ≤ x ≤ , show that ( a y .
Summary ® A real valued function is