Thank you for the questions. Here are your answers.
Question 1
----------
We differentiate
y= (x^2+2)(3x-1)
with respect to x .
We will expand the function into a polynomial and the differentiate
that polynomial:
y=(x^2+2)(3x-1)
= (x^2+2)3x + (x^2+2)(-1)
= 3x^3+6x -x^2 -2
= 3x^3 -x^2 +6x -2
Now can just differentiate this polynomial directly:
dy/dx = 9x^2 - 2x + 6
This is the answer.
Question 2
---------
In this question and the succeeding questions, we have to find an
appropriate u so that we can use the chain rule to simplify the
computation.
We differentiate
y= (2x-4)^2
with respect to x .
We set u=2x-4 .
Then y=u^2 .
Hence,
dy/du = d(u^2)/du = 2u
and
du/dx = d(2x-4)/dx = 2 .
Hence, by the chain rule:
dy/dx = (dy/du) (du/dx)
= 2u (2)
= 4u
= 4(2x-4)
= 8x-16
Question 3
----------
We differentiate
i=20 sin (100 pi t + pi/3)
with respect to t.
We let u= 100 pi t + pi/3 .
Then du/dt = 100 pi .
We have
i=20 sin (u)
so
di/du = 20 cos (u) .
Hence by the chain rule,
di/dt = (di/du) (du/dt)
= 20 cos(u) 100 pi
= 2000 cos(u)
= 2000 cos (100 pi t + pi/3)
Question 4
----------
We differentiate
i = 50 cos (50 pi t - pi/6)
with respect to t.
We set
u= 50 pi t - pi/6 .
Then
du/dt = 50 pi
We have
i = 50 cos (u)
so
di/du = -50 sin (u)
Hence by the chain rule:
di/dt= (di/du)(du/dt)
= -50 sin (u) 50 pi
= -2500 sin (u)
= -2500 sin (50 pi t - pi/6)
======================
Plots:
I have made simple plots of each function and its derivative. This
might help you to visualize what is going on with the functions and
the derivatives.
One simple thing to observe in these plots is that whenever the
function has a local maximum or minimum (that is, at the top of a
"hill" or the bottom of a "valley" in the graph of the function) the
value of the deriative is always 0 there. This is a general property
of the deriviative - it's one of the important uses of deriviatives,
actually, as it is used in physics in a lot.
The plots are at:
http://www.rbnn.com/google/differentiation/1.jpg
http://www.rbnn.com/google/differentiation/2.jpg
http://www.rbnn.com/google/differentiation/3.jpg
http://www.rbnn.com/google/differentiation/4.jpg
for questions 1 through 4 respectively. I used Matlab to generate
these plots. Matlab is an excellent program, see
http://www.mathworks.com , and I highly recommend it for playing
around with mathematical functions.
==================
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