Learn Miscellaneous Exercise 5 with Free Lessons & Tips

Question 1:

Differentiate w.r.t. x the function

Solution 1:

Comments

Harshit Bhardwaj

Question 2:

Differentiate w.r.t. x the function

Solution 2:

Using chain rule, we obtain

Comments

Harshit Bhardwaj

Question 3:

Differentiate w.r.t. x the function

Solution 3:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Harshit Bhardwaj

Question 4:

Differentiate w.r.t. x the function

Solution 4:

Using chain rule, we obtain

Comments

Harshit Bhardwaj

Question 5:

Differentiate w.r.t. x the function

Solution 5:

Comments

Harshit Bhardwaj

Question 6:

Solution 6:

Therefore, equation (1) becomes

Comments

Harshit Bhardwaj

Question 7:

Differentiate w.r.t. x the function

Solution 7:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Harshit Bhardwaj

Question 8:

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

Solution 8:

By using chain rule, we obtain

Comments

Harshit Bhardwaj

Question 9:

Differentiate w.r.t. x the function

Solution 9:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Comments

Harshit Bhardwaj

Question 10:

Differentiate w.r.t. x the function

, for some fixed and

Solution 10:

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

Comments

Harshit Bhardwaj

Question 11:

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

Solution 11:

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

Comments

Harshit Bhardwaj

Question 12:

Find, if

Solution 12:

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Harshit Bhardwaj

Question 13:

Find, if

Solution 13:

Comments

Harshit Bhardwaj

Question 14:

If, for, −1 < x <1, prove that

Solution 14:

_{It is given that,}

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

_{Hence, proved.}

Comments

Harshit Bhardwaj

Question 15:

If, for some prove that

is a constant independent of a and b.

Solution 15:

It is given that,

Differentiating both sides with respect to x, we obtain

Hence, proved.

Comments

Harshit Bhardwaj

Question 16:

If with prove that

Solution 16:

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.

Comments

Tasneem

Question 17:

If and, find

Solution 17:

Comments

Harshit Bhardwaj

Question 18:

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

Solution 18:

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,

Comments

Harshit Bhardwaj

Question 19:

Using mathematical induction prove that for all positive integers n.

Solution 19:

_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.}

Comments

Harshit Bhardwaj

Question 20:

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 20:

Differentiating both sides with respect to x, we obtain

Comments

Harshit Bhardwaj

Question 21:

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

Solution 21:

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 22:

If, prove that

Solution 22:

Thus,

Comments

Harshit Bhardwaj

Question 23:

If, show that

Solution 23:

It is given that,

View

Comments

Tasneem

All Exercises - Chapter 5 - Continuity and Differentiability

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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