Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

If e^y (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

Evaluate \(\underset { x\rightarrow -3 }{ lim } \frac { { x }^{ 3 }+27 }{ { x }^{ 5 }+243 } \)

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)