differentiation-5

4) Differentiation of a function w.r.t other functions:

Illustration:

1) Differentiate with respect to tan^-1x.

Solution:

2) Differentiate ln tanx with respect to  

Solution:

5) Higher Order derivatives:

Let y = f(x) be differentiable function such that z = f¹(x) is also differentiable. Then second derivative of y = f(x) is denoted by and .

In general

Illustration 11:

For the curve find the rate of change of slope at (4, 27).

Solution:

We know that slope of a curve is given by . Now let g(x) denote slope of curve.

So,

Now rate of change of slope =

So rate of change of slope at (4, 27)

= 24(2´4+1)

= 216.

                                                DIFFERENTIATION (EASY TYPE)

1) Differentiate with respect to x.

Solution:

Let y =  then,

2) Let f be twice differentiable such that f¹¹(x) = -f(x) and f¹(x) = g(x). Where h (5) = 11, find h (10).

Solution:

Given on differentiating both sides w.r.t x we get

Now

From (1) and (2) we get,

Therefore h¹(x) = 0.

So, h(x) must be constant [as d/dx constant = 0]

But h (5) = 11 so h (x) = 11

Hence, h (10) = 11.

3) Find the sum of series using calculus.

Solution:

Let which is a geometric progression.

Therefore =

On differentiating both sides, we get,

4) If and then find .

Solution:

Differentiating w.r.t x, we get,

5) If then find at .

Solution:

Putting t = cosq.

6) If a polynomial of degree 3 then find in terms of P(x) and its derivative.

Solution:

We have

And

Also

7) If

Solution:

Taking log on both sides

ylogx + xlogy = log1.

Differentiating on both sides we get,

8) If x = a (t+sint) and y = a (1-cost), Then find  

Solution:

Here x = a (t+sint) and y = a (1-cost).

Differentiating both sides w.r.t t we get,

Again differentiating both sides we get,