We are given the vector-valued function
r of t and asked to find the derivative.
The given vector-valued function r of t
has components x of t,
y of t,
z of t
which means r prime of t
will have components x prime to t,
y prime of t,
z prime of t.
So to find the derivative,
we need to differentiate each component
with respect to t.
Let's begin with the x component
where x of t equals three
divided by the quantity
negative five t minus seven.
We define x prime of t
which requires the chain rule,
where in this case u equals
the denominator of
negative five t minus seven
and therefore u prime is
equal to negative five.
Which means x of t is really equal
to three divided by u,
which equals three u to
the power of negative one.
And therefore x prime of t is equal
to negative three times u
raised to the power of
negative one minus one
which is negative two times u prime.
Which is equal to negative
three times the quantity
negative five t minus seven
to the power of negative two
times negative five, simplifying.
Let's move negative five t
minus seven to the denominator
which will change us out of the exponent
giving us the quantity of negative five
t minus seven squared
in the denominator.
In the numerator we have negative
three times negative five,
which is positive 15.
So for r prime of t,
we now know the x component
is 15 divided by the quantity
negative five t minus seven squared.
And now let's find y prime
of t on the next slide.
Y of t equals negative two t
divided by the quantity
three t squared plus four.
Define this derivative.
We won't be using the chain
rule, we will be using
the quotient rule shown here for review.
Let's begin with the denominator.
The denominator is a denominator squared
which is the quantity three
t squared plus 4 squared.
In the numerator we have the denominator
times the derivative of the numerator
minus the numerator times
the derivative of the denominator.
Which means here we have the quantity
three t squared plus four
times the derivative of negative two t
which is negative two.
And then minus the
numerator of negative two t
times the derivative of the denominator.
The derivative of three
t plus four is six t.
And now we simplify.
Distributing negative two gives us
negative six t squared minus eight.
And then negative two t times six t
is negative 12 t squared,
but we're subtracting
negative 12 t squared
which simplifies to plus 12 t squared.
And our final step, combining
like terms in the numerator.
Negative six t squared plus 12 t squared
is six t squared.
We have six t squared minus eight.
The denominator remains, the quantity
three t squared plus four squared.
So this is y prime of t, the
y component of r prime of t.
Let's go back to the previous slide
and record the y
component of r prime of t.
We have the quantity six
t squared minus eight
divided by the quantity three t
squared plus four squared.
And now let's find the
derivative of the z component.
Where z of t equals
negative four t squared
divided by the quantity negative t
to the third minus seven.
And once again, this is going
to require the quotient rule.
So z prime of t is equal to,
again let's start with the denominator.
The denominator is the denominator squared
so we have the quantity
at negative t cubed
minus seven squared.
And the numerator is the denominator
of negative t cubed minus seven
times the derivative of the numerator.
The derivative of
negative four t squared is
negative eight t.
And we have minus
the numerator of negative four t squared
times the derivative of the denominator.
The derivative of negative
two t cubed minus seven
is negative three t squared.
And now we simplify.
Distributing negative eight t gives us
positive eight t to the fourth
plus 56 t.
And then we have negative four t squared
times negative three t squared
that's positive 12 t to the fourth.
But we're subtracting so we have
minus 12 t to the fourth.
Simplifying one last time
by combining like terms in the numerator,
eight t to the fourth
minus 12 t to the fourth
is negative four t to the fourth.
The numerator is negative
four t to the fourth
plus 56 t.
Which is the z component of r prime of t.
And now we have r prime of t.
I hope you found this helpful.
