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Q. 1. and f '(x) = sin x^{2} , then ....................
Ans.
Solution.
Q. 2. If f_{r} (x), g (x), h_{r} (x) r , r = 1, 2, 3 are polynomials in x such that fr(a) = g_{r} (a) = h_{r}(a), r = 1, 2, 3
....................
Ans. 0
Solution.
Where f_{r}(x), g_{r}(x), h_{r}(x), r = 1, 2, 3, are polynominals in x and hence differentiable and
f_{r}(a) = gr(a) = hr(a), r = 1, 2, 3 … (2)
Differentiating eq. (1) with respect to x, we get
Using eq. (2) we get D_{1} = D_{2} = D_{3} = 0 [By the property of determinants that D = 0 if two rows in D are identical]
∴ F' (a) = 0.
Q. 3. If f(x) = log_{x} (ln x), then f '(x) at x = e is ....................
Ans. 1/e
Solution. Given that
Q. 4. The derivative of at = 1/2 is .......
Ans. 4
Solution.
Q. 5. If f (x) =  x – 2  and g(x) = f [f(x)], then g'(x) = .................... for x > 20
Ans. 1
Solution. f (x) =  x – 2 
⇒ g (x) = f (f (x)) =  f (x) – 2  as x > 20
=  x  2 2  = x  2 2 as x > 20 =  x – 4 
= x – 4 as x > 20
∴ g' (x) = 1
Q. 6. If xe^{xy} = y + sin^{2} x, then at x = 0, dy/dx = ......
Ans. 1
Solution. Given : xe^{xy} = y + sin^{2} x
Differentiating both sides w. r.to x, we get
True/ False
Q. 1. The derivative of an even function is always an odd function.
Ans. T
Solution. , which is an even function
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