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**Fill in the Blanks**

**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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