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Category Archives: calculus
Derivatives part 14: IITJEE maths tutorial problems for practice
This is part 14 of the series
Question 1:
Let for be a real valued function. Then, find for :
Answer 1:
Consider so that we have
Hence, when
Hence,
Question 2:
Let , then find .
Answer 2:
Given that ….call this I.
Also, from above, we get …call this II.
so we get ….call this I’
and …call this II’.
and hence,
Also, again ….A
…B
So, we now we get the following two equations:
…..A’
….B’
so, now we have so that we get and
so
Question 3:
If and , then find .
Answer 3:
Given that where a is a parameter (constant) and t is a variable.
Let so that
so that
so that we have
Question 4:
If then find
Answer 4:
Given that and put
so that
which is the required answer.
Question 5:
If , where a is a parameter, then find .
Answer 5:
Given that so that
Differentiating both sides w.r.t. x, we get
Question 6:
If then find .
Answer 6:
Given that so that where so that
and
Let so that
Now, note that so we get the following simplification:
Now,
Cheers,
Nalin Pithwa
Derivatives: part 13: IITJEE Math tutorial problems for practice
Question 1:
Let be a differentiable function w.r.t. x at and , then evaluate
Solution 1:
By definition, derivative of a function is , where let us substitute , , as , then
So that above expression is equal to
exists and can be evaluated if we know the value of the function at .
Question 2:
If , then find .
Answer 2:
Given that
Taking derivative of both sides w.r.t. x, we get the following equation:
This further simplifies to :
But, we already know that so that
is the desired answer.
You can see how ugly it looks. Is there any way to simplify above? Let us give it one more shot. As follows:
Given that Hence, so that
. If , then
. Taking derivative of both sides w.r.t. x, we get:
which is such an elegant answer ðŸ™‚
Question 3:
If , where c is a parameter constant, then find at .
Solution 3:
Let and .
Taking logarithm of both sides:
and .
Consider the LHS equation:
Taking derivative of both sides w.r.t.x, we get:
.
Also,
Now substitute and get the required answer.
Substituting
Hence, then, is the desired answer.
Question 4:
Find
Answer 4:
Consider
Subsituting and , we get the following:
which in turn equals noting that
Hence, the answer is
Question 5:
If , find
Solution 5:
Given that
. Taking derivative of both sides w.r.t. x,
which in turn equals
But,
so that
Hence,
Hence,
Question 6:
Find
Solution 6:
Let
Put so that
where
Question 7:
If . Find .
Solution 7:
Let so that
so that
We now have
so that
so the desired answer is
Question 8:
If then find
Solution 8:
Given that
but and
so now we have
Hence, we get .
Question 9:
If and and , then evaluate .
Solution 9:
We have and hence,
By definition of derivative, we have where let us say so that , and
and and hence, .
Hence,
Question 10:
If , and , then find .
Answer 10:
Let and hence,
Hence,
so that
Let so that and
Hence, we get so that
Hence,
Hence,
Hence, and hence and so hence,
so that
….call this A.
…call this B.
which in turn equals
so where did we go wrong….quite clearly, practice alone can help us develop foresight…below is a cute proof:
and put
so that so we have bingo ðŸ™‚ an elegant answer
Cheers,
Nalin Pithwa.
Derivatives: part 12:IITJEE maths tutorial problems for practice
1, , , then find .
Option (A)
Option (B)
Option (C)
Option (D)
Solution 1: Given that so that
and given that so that
and so we get so that correct choice is option A.
2. If and then find when
Option (A) 0
Option (B)
Option (C)
Option (D)
Solution 2:
which is equal to the following at
so that the correct choice is C.
3. If , then find
Option A:
Option B:
Option C:
Option D:
Solution 3:
Let where we put so now let
So, we get and and
So we get
So now
And,
Hence, we get the following:
Question 4: Find the following:
Option a:
Option b:
Option c:
Option d:
Solution 4:
Let
Let , , , and
so
Let
Let , and
Let
so that
so that so the option is a.
Question 5:
If then find .
Solution 5:
Question 6:
If , then is equal to
(a) 1/2 (b) 1/3 (c) 1/6 (d) 0
Solution 6:
Given that
Hence, we have
So, at x=0, on substitution we get .
Question 7:
If , , then find .
Solution 7:
Given , let so that
so that
Now, so that
so now
.
Question 8:
Find
Solution 8:
Let it be given that
Now, let us simplify this as where and
Now, first consider . Taking derivative of both sides w.r.t. x, we get
….A
Now, next consider . Takind derivative of both sides w.r.t. x, we get
….B
So that we get using A and B.
Question 9:
If , then find
Solution 9:
Given that
. Taking derivative of both sides w.r.t. x, we get
which is the required answer.
Question 10:
If , then find .
Solution 10:
Given that
Taking logarithm of both sides w.r.t. any arbitrary valid base,
so that
Taking derivative of both sides w.r.t. x, we get the following:
, so that finally we get the desired answer:
More later,
Cheers,
Nalin Pithwa
Derivatives: part 11: IITJEE maths tutorial problems for practice
Problem 1: Find .
Choose (a) (b) (c) (d)
Solution 1:
Let . Hence, . Differentiating both sides w.r.t. x, we get the following:
But,
Hence, the answer is . Option c.
Problem 2: Find if
Choose (a) (b) (c) (d)
Solution 2:
The given equation is . Differentiating both sides wrt x,
is the answer. Option D.
Problem 3: If then is
choose (a) (b) (c) (d)
Solution 3:
Given that so that we have
so now differentiating both sides w.r.t. x,
Now, we also know that
But, note that by laws of logarithms, on simplification, we get
and so that on squaring, we get
so that now we get
, which all put together simplifies to
so that the answer is option C.
Problem 4: Find
Choose option (a) (b) (c) (d)
Solution 4:
Let us consider the first differential. Let us substitute . Hence,
and so we , and so also, we get so we get
required derivative
. Answer is option C.
Problem 5: Find
Choose option (a) zero (b) 26 (c) 26! (d) does not exist
Solution 5: the expression also includes a term so that the final answer is zero only.
Problem 6: Find .
Solution 6: Let
so
so so that differentiating both sides w.r.t. x, we get
we get
we get
we get
so the answer is option B.
Choose option (a): (b) (c) (d) none of these
Problem 7:
Find
Choose option (a) (b) (c) (d)
Solution 7: Let so that taking logarithm of both sides
so that
. Differentiating both sides w.r.t.x we get:
so that we get now
$latex\frac{1}{y(\log{y})} \times \frac{dy}{dx} = 1 + \log{x} $
so we get option a as the answer.
Problem 8:
Find
Choose option (a): (b) (c) (d) none of these.
Solution 8:
let taking logarithm of both sides we get
and now differentiating both sides w.r.t.x, we get
and now let and again take logarithm of both sides so that we get (this is quite a classic example…worth memorizing and applying wherever it arises):
The answer is option C.
Problem 9:
Find .
Choose option (a): (b) (c) (d) none of these
Solution 9:
Given that
Remark: Simply multpilying out thinking the symmetry will simplify itself is going to lead to a mess…because there will be no cancellation of terms …:) The way out is a simple algebra observation…this is why we should never ever forget the fundamentals of our foundation math:)
note that the above can be re written as follows:
Now, we are happy like little children because many terms cancel out ðŸ™‚ hahaha…lol ðŸ™‚
and now differentiating both sides w.r.t.x we get
The answer is option A.
Problem 10:
If and then find the value of at
Choose option (a): (b) (c) (d)
Solution 10:
Answer is option D.
Regards,
Nalin Pithwa.
Derivatives: Part 10: IITJEE maths tutorial problems for practice
Problem 1: If , and , then is equal to:
(a) (b) (c) (d)
Problem 2: If , and , then is equal to:
(a) (b) (c) (d)
Problem 3: is equal to:
(a) (b) (c) cosec(x) (d)
Problem 4: , then is:
(a) (b) (c) (d)
Problem 5: is equal to:
(a) (b) (c) (d)
Problem 6: then is equal to :
(a) (b) (c) (d)
Problem 7: If then is
(a) (b) (c) (d)
Problem 8: is:
(a) (b)
(c) (d)
Problem 9: If then is:
(a) (b) 0 (c) 1 (d)
Problem 10: is:
(a) (b) (c) (d)
Regards,
Nalin Pithwa.
Derivatives: Part 9: IITJEE maths tutorial problems practice
Problem 1: is equal to:
(a) (b)
(c) (d)
Problem 2: is equal to:
(a) (b) (c) (d)
Problem 3: If where , then is given by :
(a) (b) (c) (d)
Problem 4: is equal to:
(a) (b) (c) (d)
Problem 5:
If , and , then is equal to:
(a) (b) (c) (d)
Problem 6: is equal to:
(a) (b) (c) (d)
Problem 7: is equal to
(a) 0 (b) (c) (d)
Problem 8: If , then is equal to:
(a) (b) (c) (d)
Problem 9: is equal to:
(a) (b) (c) 9d)
Problem 10: If then is equal to:
(a) (b) (c) (d)
Cheers,
Nalin Pithwa
Derivatives: part 8: IITJEE mains tutorial problems practice
Problem 1: If , then is equal to:
(a) (b) 1 (c) 0 (d) b
Problem 2: If , then is:
(a) (b) (c) (d)
Problem 3: is equal to:
(a) (b) (c) (d)
Problem 4: If , then is equal to:
(a) (b) (c) (d)
Problem 5: is equal to:
(a) (b) (c) (d)
Problem 6: If then the value of is
(a) (b) (c) (d)
Problem 7: is equal to:
(a) (b) (c) (d) none
Problem 8: is equal to:
(a) (b)
(c) (d)
Problem 9: If and , then is :
(a) (b) (c) (d)
Problem 10: is equal to:
(a) (b) (c) (d)
Cheers,
Nalin Pithwa.
Derivatives: part 7: IITJEE tutorial problems practice
Problem 1: Differential coefficient of is
(a) (b) (c) (d)
Problem 2: If , then is equal to :
(a) 2 (b) 2 (c) 1 (d) 1
Problem 3: If , then is equal to:
(a) (b)
(c) (d) zero
Problem 4: For the differentiable function f, the value of : is equal to:
(a) (b) (c) (d) zero
Problem 5: The derivative of w.r.t. at is :
(a) (b) (c) (d) 1
Problem 6: If then is
(a) (b) (c) (d)
Problem 7: Consider the following statements:
(1) (2)
(3) (4)
Which of the following statements are true?
(a) 1 and 2 (b) 2 and 3 (c) 2 and 4 (d) 3 and 4
Problem 8: If then
(a) (b) (c) (d)
Problem 9: If , then find the value of is:
(a) (b) (c) (d)
Problem 10: If then the value of is:
(a) (b) (c) (d)
Cheers,
Nalin Pithwa.