If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
012. The Chain Rule
Question: `qNote that there are 12 questions in this assignment.
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Question: `q001. When we form the composite of two functions, we first apply one function to the variable, then we apply the other function to the result.
We can for example first apply the function z = t^2 to the variable t, then we can apply the function y = e^z to our result.
If we apply the functions as specified to the values t = -2, -1, -.5, 0, .5, 1, 2, what y values to we get?
Your solution:
Confidence Assessment:
Given Solution:
`aIf t = -2, then z = t^2 = (-2)^2 = 4, so y = e^z = e^4 = 55, approx..
If t = -1, then z = t^2 = (-1 )^2 = 1, so y = e^z = e^1 = 2.72, approx..
If t = -.5, then z = t^2 = (-.5 )^2 = .25, so y = e^z = e^.25 = 1.28, approx.
If t = 0, then z = t^2 = ( 0 )^2 = 0, so y = e^z = e^0 = 1.
If t = .5, then z = t^2 = (.5 )^2 = .25, so y = e^z = e^.25 = 1.28, approx.
If t = 1, then z = t^2 = ( 1 )^2 = 1, so y = e^z = e^1 = 2.72, approx..
If t = 2, then z = t^2 = ( 2 )^2 = 4, so y = e^z = e^4 = 55, approx..
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Question: `q002. If we evaluate the function y = e^(t^2) at t = -2, -1, -.5, 0, .5, 1, 2, what y values do we get?
Your solution:
Confidence Assessment:
Given Solution:
`aIf t = -2, then y = e^(t^2) = e^(-2)^2 = e^4 = 55, approx.
If t = -1, then y = e^(t^2) = e^(-1)^2 = e^1 = 2.72, approx.
If t = -.5, then y = e^(t^2) = e^(-.5)^2 = e^.25 = 1.28, approx.
If t = 0, then y = e^(t^2) = e^(0)^2 = e^0 = 1.
If t = .5, then y = e^(t^2) = e^(.5)^2 = e^.25 = 1.28, approx.
If t = 1, then y = e^(t^2) = e^(1)^2 = e^1 = 2.72, approx.
If t = 2, then y = e^(t^2) = e^(2)^2 = e^2 = 55, approx.
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Question: `q003. We see from the preceding two examples that that the function y = e^(t^2) results from the 'chain' of simple functions z = t^2 and y = e^z.
What would be the 'chain' of simple functions for the function y = cos ( ln(x) )?
Your solution:
Confidence Assessment:
Given Solution:
`aThe first function encountered by the variable x is the ln(x) function, so we will say that z = ln(x). The result of this calculation is then entered into the cosine function, so we say that y = cos(z).
Thus we have y = cos(z) = cos( ln(x) ).
We also say that the function y(x) is the composite of the functions cosine and natural log functions, i.e., the composite of y = cos(z) and z = ln(x).
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Question: `q004. What would be the chain of functions for y = ( ln(t) ) ^ 2?
Your solution:
Confidence Assessment:
Given Solution:
`aThe first function encountered by the variable t is ln(t), so we say that z = ln(t). This value is then squared so we say that y = z^2.
Thus we have y = z^2 = (ln(t))^2.
We also say that we have here the composite of the squaring function and the natural log function, i.e., the composite of y = z^2 and z = ln(t).
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Question: `q005. What would be the chain of functions for y = ln ( cos(x) )?
Your solution:
Confidence Assessment:
Given Solution:
`aThe first function encountered by the variable is cos(x), so we say that z = cos(x). We then take the natural log of this function, so we say that y = ln(z).
Thus we have y = ln(z) = ln(cos(x)).
This function is the composite of the natural log and cosine function, i.e., the composite of y = ln(z) and z = cos(x).
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Question: `q006. The rule for the derivative of a chain of functions is as follows:
The derivative of the function y = f ( g ( x) ) is y ' = g' ( x ) * f ' ( g ( x) ).
For example if y = cos ( x^2 ) then we see that the f function is f(z) = cos(z) and the g function is the x^2 function, so that f ( g ( x) ) = f ( x^2 ) = cos ( x^2 ) . By the chain rule the derivative of this function will be
(cos(x^2)) ' = g ' ( x) * f ' ( g ( x) ) .
g(x) = x^2 so g'(x) = 2 x.
f ( z ) = cos ( z) so f ' ( z ) = - sin( z ), so f ' ( g ( x ) ) = - sin ( g ( x ) ) = - sin ( x^2).
Thus we obtain the derivative
(cos(x^2)) ' = g ' ( x ) * f ' ( g ( x ) ) =
2 x * ( - sin ( x^2 ) ) =
- 2 x sin ( x^2).
Apply the rule to find the derivative of y = sin ( ln ( x ) ) .
Your solution:
Confidence Assessment:
Given Solution:
`aWe see that y = sin ( ln(x) ) is the composite f(g(x)) of f(z) = sin(z) and g(x) = ln(x). The derivative of this composite is g ' (x) * f ' ( g(x) ).
Since g(x) = ln(x), we have g ' (x) = ( ln(x) ) ' = 1/x.
Since f(z) = sin(z) we have f ' (z) = cos(z).
Thus the derivative of y = sin( ln (x) ) is
y ' = g ' (x) * f ' (g(x)) = 1 / x * cos ( g(x) ) =
1 / x * cos( ln(x) ).
Note how the derivative of the 'inner function' g(x) = ln(x) appears 'out in front' of the derivative of the 'outer' function; that derivative is in this case the sine function.
STUDENT QUESTION
We have been given many different
derivative functions in the last two assignments are these things that will
always be
given or should I make a chart of them and start learning them by memory.
INSTRUCTOR RESPONSE
There are not that many basic functions (power, exponential, logarithmic, sine, cosine cover most of it). You will need to know the derivatives of these functions and maybe a few more as we go along, and the rules for derivatives of product, quotient and composite functions.
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Question: `q007. Find the derivative of y = ln ( 5 x^7 ) .
Your solution:
Confidence Assessment:
Given Solution:
`aFor this composite we have f(z) = ln(z) and g(x) = 5 x^7. Thus
f ' (z) = 1 / z and g ' (x) = 35 x^6.
We see that f ' (g(x)) = 1 / g(x) = 1 / ( 5 x^7).
So the derivative of y = ln( 5 x^7) is
y ' = g ' (x) * f ' (g(x)) = 35 x^6 * [ 1 / ( 5 x^7 ) ].
Note again how the derivative of the 'inner function' g(x) = 5 x^7 appears 'out in front' of the derivative of the 'outer' function; that derivative is in this case the reciprocal or 1 / z function.
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Question: `q008. Find the derivative of y = e ^ ( t ^ 2 ).
Your solution:
Confidence Assessment:
Given Solution:
`aThis function is the composite of f(z) = e^z and g(t) = t^2. We see right away that f ' (z) = e^z and g ' (t) = 2t.
Thus the derivative of y = e^(t^2) is y ' = g ' (t) * f ' (g(t)) = 2 t * e^(t^2).
Note once more how the derivative of the 'inner function' g(t) = t^2 appears 'out in front' of the derivative of the 'outer' function.
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Question: `q009. Find the derivative of y = cos ( e^t ).
Your solution:
Confidence Assessment:
Given Solution:
`aWe have the composite of f(z) = cos(z) and z(t) = e^t, with derivatives f ' (z) = -sin(z) and g ' (t) = e^t.
Thus the derivative of y = cos ( e^t ) is y ' = g ' (t) * f ' (g(t)) = e^t * -sin( e^t) = - e^t sin ( e^t).
Note how the 'inner function' is unchanged, as it has been in previous examples, and how its derivative appears in front of the derivative of the 'outer' function.
STUDENT QUESTION: I am confused as to why the end result is - e^t sin (e^t). Is it because we just combined the multiplication?
<h3>I believe you are asking why e^t * -sin( e^t) = - e^t sin ( e^t)
e^t * (-sin( e^t)) = (- sin ( e^t) ) * e^t by the commutativity of
multiplication
- sin ( e^t) * e^t = - ( sin ( e^t) * e^t) by the laws for multiplying signed
numbers.
- ( sin ( e^t) * e^t) = - ( e^t * sin(e^t) ) by commutativity
- ( e^t * sin(e^t) ) = - e^t * sin(e^t) by the laws for multiplying signed
numbers
So e^t * -sin( e^t) = - e^t sin ( e^t).</h3>
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Question: `q010. Find the derivative of y = ( ln ( t ) ) ^ 9.
Your solution:
Confidence Assessment:
Given Solution:
`aWe have y = f(g(t)) with f(z) = z^9 and g(t) = ln(t). f ' (z) = 9 z^8 and g ' (t) = 1 / t. Thus
y ' = g ' (t) * f ' (g(t)) = 1/t * 9 ( ln(t) )^8.
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Question: `q011. Find the derivative of y = sin^4 ( x ).
Your solution:
Confidence Assessment:
Given Solution:
`aThe composite here is y = f(g(x)) with f(z) = z^4 and g(x) = sin(x). Note that the notation sin^4 means to raise the value of the sine function to the fourth power.
We see that f ' (z) = 4 z^3 and g ' (x) = cos(x). Thus
y ' = g ' (x) * f ' (g(x)) = cos(x) * 4 ( sin(x) ) ^ 3 = 4 cos(x) sin^3 (x).
STUDENT QUESTION: I am having a hard time figuring out the simplication of this one? Any help?
INSTRUCTOR RESPONSE: You had the right function for the f and g functions, except that you reversed them:
If f(z) = sin(z) and g(x) = x^4, then f(g(x)) = sin(x^4),
and the derivative would indeed be 4 x^3 * cos(x^4).
However the function here is sin^4(x), so f(z) = z^4 and g(x) = sin(x). This
gives a very different result, as shown in the given solution.
Regarding the simplification step cos(x) * 4 ( sin(x) ) ^ 3 = 4 cos(x) sin^3 (x):
a * b * c = b * a * c; this actually requires multiple applications of associative and commutative laws for addition, but it's something we're used to so we don't usually think about it that deeply. We know it is 'safe' to change the order of a sequence of multiplications.
cos(x) * 4 ( sin(x) ) ^ 3 is just a sequence of multiplications. If a = cos(x), b = 4 and c = (sin(x))^3, then this expression is a * b * c.
Then b * a * c = 4 cos(x) sin^3 (x).
Note that sin^3(x) means (sin(x))^3.
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Question: `q012. Find the derivative of y = cos ( 3x ).
Your solution:
Confidence Assessment:
Given Solution:
`aThis is the composite y = f(g(x)) with f(z) = cos(z) and g(x) = 3x. We obtain f ' (z) = - sin(z) and g ' (x) = 3. Thus
y ' = g ' (x) * f ' (g(x)) = 3 * (-sin (3x) ) = - 3 sin(3x).
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