We want to determine the equation
of the tangent line to y equals x times the cube root of x
and parallel to y equals 4x minus three.
Remember if two lines are parallel,
then these slopes are the same.
So in this case,
we know the tangent line must have a slope of four.
So what we'll have to do is determine the point
of tangency by determining where the first derivative
of the given function is equal to four.
And then we'll have to determine the y coordinate
on the original function
so that we can then determine the equation
of the tangent line.
So first, the given function or equation is not
in the form where we can find the first derivative,
because we have the cube root here.
For the cube root, the index is two
and the exponent on x is one.
And therefore, y equals x times the cube root of x
is equal to x times x to the power of 1/3.
And now we can multiply the exponent on x is one,
the basis of the statement we're multiplying
and therefore we add the exponents.
This gives us x the power of one plus 1/3
or 3/3 plus 1/3 which is 4/3.
So we're going to be using the function in the form
of y equals x of the 4/3 to find the derivative.
So let's find y prime, the first derivative.
Applying the power rule of differentiation,
the derivative of x
to the 4/3 power with respect to x is equal
to 4/3 times x to the power of 4/3 minus one
which gives us 4/3 times x to the power of 1/3.
Now that we have the derivative function,
because the tangent line must be parallel
to y equals 4x minus three,
the line y equals 4x minus three has a slope
of positive four because it's in sloped intercept form,
where m is positive four.
So what we'll do is set the first derivative equal to four
and then solve for x.
This will give us where the tangent line
will have a slope of four
and therefore it'll be parallel to this given line.
So now we're going to set 4/3 x
to the power of 1/3, which is a derivative
equal to the slope of four and solve for x.
First, we need to isolate x to the power of 1/3,
by multiplying both sides of the equation
by the reciprocal of 4/3, which is 3/4.
On the left because the fraction's reciprocals,
the product is one which gives us
x to the power of 1/3 equals on the right, notice,
the fours simplify to one, leaving us with just three.
So now to solve for x, we need to have x to the first power
not x to the 1/3 power.
X to the 1/3 power is equivalent to the cube root of x
to get an exponent of one or undo the cube root,
we will now cube both sides of the equation
or raise both sides of the equation to the third power.
Because we have powers, the powers here on the left.
We multiply the exponents.
1/3 times three is one giving us x to the first
or x equals on the right, three cubed is 27.
Again, this equation here is equivalent
to having the cube root of x cubed
equals three cubed,
which still gives us x equals 27.
So now we know the point of tangency,
we'll call it point p has an x coordinate of 27
and the y coordinate is going to be the value we get
from the function, when we substitute 27 for x,
which we'll call y of 27.
And we also know the slope of a tangent line is the same
as a slope of a given line, which is four.
So let's go ahead and state here
the slope m is equal to four.
And now we need to find y of 27.
Let's go ahead and do that.
Y of 27 is equal to 27 times the cube root of 27.
The cube root of 27 is equal to three
since three cubed equals 27,
giving us 27 times three, which is equal to 81.
So again now we know the point of tangency is 27, 81
and the slope of the tangent line is four.
We have all the information we need
to find the equation of of the tangent line.
Let's go ahead and use point slope form first
and then we'll solve for y to write the equation
in sloped intercept form.
For point slope form x one is 27,
y one is 81, and of course m is four.
The equation of the tangent line in point slope form
is y minus 81 equals four times the quantity x minus 27.
This is the equation of the tangent line
to the given function that is parallel to the line
y equals four x minus three.
This is the point slope form of the line,
but let's also solve for y and write the equation,
sloped intercept form,
and therefore we'll distribute four
and then add 81 to both sides.
So distributing four, we have y minus 81 equals
four x minus 108, adding 81 to both sides
of the equation and simplifying.
We have y equals four x negative 108 plus 81
is equal to negative 27, giving us minus 27.
Let's verify this graphically.
The graph of y equals x times the cube root of x
is graphed in blue.
The line y equals four x minus three, is this green line.
We found the point of tangency to be the location,
where the tangent line would be parallel to the green line.
We found the point 27, 81, this red point here
and the equation of y equals four x minus 27
is this red line, which we can see is tangent to the graph
at the point, and also parallel to the given green line.
The graph does verify our work is correct.
I hope you found this helpful.