# differentiation-5

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

_{ }

Illustration:

1) Differentiate _{ } with respect to tan^{-1}x.

Solution:

_{ }

2) Differentiate ln tanx with respect to _{ }

Solution:

_{ }

5) Higher Order derivatives:

Let y = f(x) be differentiable function such that z = f^{1}(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 f^{11}(x) = -f(x) and f^{1}(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 h^{1}(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,

_{ }