21.22 - More Miscellaneous Examples

In the previous section, we saw some miscellaneous examples. In this section, we will see a few more examples.

Solved example 21.68
Find f'(x) if f(x) = $\rm{(\sin x)^{\sin x}}$ for all 0<x<π.
Solution:

 

Solved example 21.69
For a positive constant a, find $\rm{\frac{dy}{dx}}$ where:
$\rm{y = a^{t + 1/t}~\text{and}~x = \left(t + 1/t \right)^a}$
Solution:
1. First we will find dy/dt:

2. Next we will find dx/dt

3. Now we can find dy/dx

 

Solved example 21.70
Differentiate $\rm{\sin^2 x~\text{w.r.t}~e^{\cos x}}$.
Solution:
1. Let $\rm{y = \sin^2 x~\text{and}~x = e^{\cos x}}$
2. We want the derivative of
$\rm{\sin^2 x~\text{w.r.t}~e^{\cos x}}$
• That is., we want the derivative of
$\rm{\sin^2 x~\text{w.r.t}~u}$
• That is., we want the derivative of
$\rm{y~\text{w.r.t}~u}$
• That is., we want $\rm{\frac{dy}{du}}$
3. First we will find dy/dx:
$\rm{y = \sin^2 x}$
So we have:$\rm{\frac{dy}{dx}~=~2 \sin x \cos x}$

4. Next we will find du/dx:

 

5. Now we can find dy/du:

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The link below gives a few more miscellaneous examples:

Miscellaneous Exercise

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In the next chapter, we will see application of derivatives.

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