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 🙂

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 🙂 an elegant answer

Cheers,

Nalin Pithwa.