Learn Miscellaneous Exercise 5 with Free Lessons & Tips

Question 1:

Differentiate w.r.t. x the function

Solution 1:

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Harshit Bhardwaj

Question 2:

If, show that

Solution 2:

It is given that,

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Tasneem

Question 3:

Differentiate w.r.t. x the function

Solution 3:

Using chain rule, we obtain

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Harshit Bhardwaj

Question 4:

Differentiate w.r.t. x the function

Solution 4:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

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Harshit Bhardwaj

Question 5:

Differentiate w.r.t. x the function

Solution 5:

Using chain rule, we obtain

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Harshit Bhardwaj

Question 6:

Differentiate w.r.t. x the function

Solution 6:

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Harshit Bhardwaj

Question 7:

Solution 7:

Therefore, equation (1) becomes

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Harshit Bhardwaj

Question 8:

Differentiate w.r.t. x the function

Solution 8:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

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Harshit Bhardwaj

Question 9:

Differentiate w.r.t. x the function , for some constant a and b.

Solution 9:

By using chain rule, we obtain

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Harshit Bhardwaj

Question 10:

Differentiate w.r.t. x the function

Solution 10:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

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Harshit Bhardwaj

Question 11:

Differentiate w.r.t. x the function

, for some fixed and

Solution 11:

Differentiating both sides with respect to x, we obtain

s = a^a

Since a is constant, a^a is also a constant.

∴

From (1), (2), (3), (4), and (5), we obtain

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Harshit Bhardwaj

Question 12:

Differentiate w.r.t. x the function , for

Solution 12:

Differentiating both sides with respect to x, we obtain

Differentiating with respect to x, we obtain

Also,

Differentiating both sides with respect to x, we obtain

Substituting the expressions of in equation (1), we obtain

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Harshit Bhardwaj

Question 13:

Find, if

Solution 13:

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Harshit Bhardwaj

Question 14:

Find, if

Solution 14:

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Harshit Bhardwaj

Question 15:

If, for, −1 < x <1, prove that

Solution 15:

_{It is given that,}

_{Differentiating both sides with respect to}_x_{, we obtain}

_{Hence, proved.}

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Harshit Bhardwaj

Question 16:

If, for some prove that

is a constant independent of a and b.

Solution 16:

It is given that,

Differentiating both sides with respect to x, we obtain

Hence, proved.

Comments

Harshit Bhardwaj

Question 17:

If with prove that

Solution 17:

Then, equation (1) reduces to

⇒sin(a+y−y) $\frac{dy}{dx}=cos^{^{2}}\left ( a+y \right )$

$\frac{dy}{dx}=\frac{cos^{2}(a+y)}{sina}$

Hence, proved.

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Tasneem

Question 18:

If and, find

Solution 18:

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Harshit Bhardwaj

Question 19:

If, show that exists for all real x, and find it.

Solution 19:

It is known that,

Therefore, when x ≥ 0,

In this case, and hence,

When x < 0,

In this case, and hence,

Thus, for, exists for all real x and is given by,

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Harshit Bhardwaj

Question 20:

Using mathematical induction prove that for all positive integers n.

Solution 20:

_For_n_{= 1,}

∴P(n) is true for n = 1

Let P(k) is true for some positive integer k.

That is,

It has to be proved that P(k + 1) is also true.

Thus, P(k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, the statement P(n) is true for every positive integer n.

_{Hence, proved.}

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Harshit Bhardwaj

Question 21:

Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Solution 21:

Differentiating both sides with respect to x, we obtain

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Harshit Bhardwaj

Question 22:

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

Solution 22:

It can be seen from the above graph that, the given function is continuos everywhere but not differentiable at exactly two points which are 0 and 1.

Comments

Tasneem

Question 23:

If, prove that

Solution 23:

Thus,

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Harshit Bhardwaj

All Exercises - Chapter 5 - Continuity and Differentiability

Exercise 5.1

Exercise 5.2

Exercise 5.3

Exercise 5.4

Exercise 5.5

Exercise 5.6

Exercise 5.7

Exercise 5.8

Miscellaneous Exercise 5

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Learn Miscellaneous Exercise 5 with Free Lessons & Tips

All Exercises - Chapter 5 - Continuity and Differentiability

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