Important Questions
4. Find 
if x = 3 cos t - 2 cos3 t, y = 3 sin t - 2 sin3 t.
Sol: 
= -3 sin t + 6 cos2 t sin t
= 3 sin t (-1 + 2 cos2 t)
= 3 sin t cos 2t.
= 3 cos t - 6 sin2 t cos t
= 3 cos t (1-2 sin2 t)
= 3 cos t cos 2t
= cot t.
5. Find the derivative of the function x = log (1+ sin2y) w.r.t. x.
Sol:
Alternate Method
x = log (1 + sin2y)
Differentiate w.r.t. y 
Short Answer Type Questions 4 Marks
1. Find the derivative of x sin x w.r.t. x from first principles.
Sol: Let f(x) = x sin x
2. If sin y = x sin (a+ y) then show that 
, (a is not a multiple of π)
Sol: sin y = x sin (a + y)
Diff. w.r.t. 'x' on both sides
Sol: xy = e x-y
taking log on both sides
y log x = x - y
4. Find the derivative of sin-1
Sol: Let y = sin-1
Let 2x = tan θ
5. If x = a (t - sin t), y = a (1 + cos t) find 
Sol: 
= a (1 - cos t) 
= a (0 - sin t)
6. If ax2 + 2hxy + by2 = 1 then prove that 
Sol: ax2 + 2hxy + by2 = 1 Diff. x on both sides
(using (1)
7 Marks Long Answer type Question
1. Find the derivative of 
a, b > 0
Sol: Let y = 
2. If 
then Show that 
Sol:
Diff. w.r.t. 'x' on both sides
3. If 
then 
Sol:
Let x = tan θ 
4. Find the derivative of 
Sol: Let U = (sinx) logx and V = x sinx
Taking log on both sides
log U = log x. log (sinx)
diff. w.r.t. x on both sides
Taking log on both sides
log V = sin x. log x
diff. w.r.t. 'x' on both sides
Now y = U + V
5. If xy + yx = ab then show that 
Sol: Let U = xy and V = yx
Taking log on both sides
log U = y logx
Diff. w.r.t. 'x' on both sides
Taking log on both sides
log V = x logy
Diff. w.r.t. 'x' on both sides
Now, xy + yx = ab U+V = ab
Writer S. V. Sailaja
