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 🙂 hahaha…lol 🙂
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.