### Video Transcript

Find the equation of the tangent to
nine π¦ squared equals negative seven π₯ plus nine that has gradient seven over
18.

So weβve got to find the equation
of the tangent, and weβre given the slope as seven over 18. So, weβre going to want to use the
equation π¦ minus π¦ nought equals π multiplied by π₯ minus π₯ nought, where this
line goes through π₯ nought, π¦ nought, with slope π. Firstly, weβll find the point at
which the line is tangent to the curve. So, that will be π₯ nought, π¦
nought. And secondly, we find the point
slope equation of the tangent line. So, we need to find π₯ nought and
π¦ nought. And the way weβll do this is by
differentiating the equation.

So weβre going to need to do this
implicitly with respect to π₯. So, we want to differentiate both
sides with respect to π₯. Now, for the right-hand side, we
recall that the derivative of ππ₯ is just π, and constants differentiate to
zero. And so, negative seven π₯ plus nine
differentiates to negative seven. The derivative of the left-hand
side is a bit trickier. This is because weβre
differentiating this function of π¦, with respect to π₯. So, we have to apply the chain
rule.

The chain rule says that dπ by dπ₯
is equal to dπ by dπ¦ multiplied by dπ¦ by dπ₯. The function we want to
differentiate is nine π¦ squared. So, if we let π equal nine π¦
squared, and so dπ by dπ₯ equals dπ by dπ¦, and this is nine π¦ squared
differentiated with respect to π¦. We can do this because we know the
power rule of differentiation. So, dπ by dπ¦ is 18π¦. And thatβs multiplied by dπ¦ by
dπ₯. And now we can put this back into
our working out.

So, we have the 18π¦ multiplied by
dπ¦ by dπ₯ equals negative seven. So now, we have an equation
involving dπ¦ by dπ₯. So, letβs rearrange to make dπ¦ by
dπ₯ the subject. We divide both sides by 18π¦. And we get dπ¦ by dπ₯ equals
negative seven over 18π¦.

Now remember, in the question we
were told that the slope dπ¦ by dπ₯ is equal to seven over 18. So, replacing dπ¦ by dπ₯ by seven
over 18, we have that seven over 18 equals negative seven over 18π¦. And, solving for π¦ gives us that
π¦ equals negative one. Substituting this back into the
original equation, nine π¦ squared equals negative seven π₯ plus nine, gives us nine
negative one squared equals negative seven π₯ plus nine. Remember that negative one squared
is just one. And then solving for π₯, we find
that π₯ equals zero.

Remember that these values that we
found are the coordinates for the point at which the line is tangent to the
curve. So, this is our π₯ nought, π¦
nought. And now, we need to use the point
slope equation to find the tangent line.

Applying the point slope equation
with π₯ nought equal to zero. π¦ nought equal to negative
one. And the slope π equal to seven
over 18. We have that the tangent line is π¦
minus negative one equals seven over 18 multiplied by π₯ minus zero. π¦ minus negative one is just π¦
plus one. And, expanding the brackets on the
right-hand side gives us that π¦ plus one equals seven over 18π₯. We can gather all of our terms onto
one side of the equation. And, we can multiply all our terms
by 18 to get rid of the fraction to give us 18π¦ minus seven π₯ plus 18 equals
zero. And, that gives us our final
answer.