Video Transcript
Find the equation of the tangent to
the curve four 𝑥 squared 𝑦 squared minus four 𝑥 minus seven 𝑦 minus one equals
zero at the point negative one, one.
In order to find the equation of a
tangent to a curve, we need to know two things: the slope of the tangent and the
coordinates of a point that lies on it. We’ve been given the coordinates of
the point negative one, one. To find the slope of the tangent,
we first need to determine the slope function of the curve, which we can do using
differentiation. As the equation of the curve has
been given implicitly, we’re going to need to use implicit differentiation to find
its slope function.
This is an application of the chain
rule. We recall that in order to find the
derivative with respect to 𝑥 of a function of 𝑦, we find the derivative with
respect to 𝑦 of that function and multiply by d𝑦 by d𝑥. As the first term in the equation
of the curve is a product of 𝑥 squared and 𝑦 squared, we also need to recall the
product rule, which states that the derivative with respect to 𝑥 of the product of
two differentiable functions 𝑢 and 𝑣 is equal to 𝑢 multiplied by d𝑣 by d𝑥 plus
𝑣 multiplied by d𝑢 by d𝑥. Essentially, we multiply each term
in the product by the derivative of the other and add them together.
Let’s begin then by finding the
derivative with respect to 𝑥 of the first term, four 𝑥 squared 𝑦 squared. By the product rule, this is equal
to four 𝑥 squared multiplied by the derivative with respect to 𝑥 of 𝑦 squared
plus 𝑦 squared multiplied by the derivative with respect to 𝑥 of four 𝑥
squared. Differentiating 𝑦 squared
implicitly with respect to 𝑥 gives two 𝑦 d𝑦 by d𝑥, and then differentiating four
𝑥 squared with respect to 𝑥 gives eight 𝑥. The entire derivative for the first
term therefore simplifies to eight 𝑥 squared 𝑦 d𝑦 by d𝑥 plus eight 𝑥𝑦
squared.
Differentiating the next term in
the equation with respect to 𝑥 gives negative four, and then differentiating
negative seven 𝑦 implicitly with respect to 𝑥 gives negative seven d𝑦 by d𝑥. Finally, differentiating the
constants negative one and zero with respect to 𝑥 gives zero, so we have the
equation eight 𝑥 squared 𝑦 d𝑦 by d𝑥 plus eight 𝑥𝑦 squared minus four minus
seven d𝑦 by d𝑥 equals zero.
We need to make d𝑦 by d𝑥 the
subject of this equation. So, we begin by collecting all
terms that don’t involve d𝑦 by d𝑥 on the right-hand side of the equation, giving
eight 𝑥 squared 𝑦 d𝑦 by d𝑥 minus seven d𝑦 by d𝑥 equals four minus eight 𝑥𝑦
squared. We then factor the left-hand side
to give eight 𝑥 squared 𝑦 minus seven multiplied by d𝑦 by d𝑥. Finally, dividing both sides of the
equation by eight 𝑥 squared 𝑦 minus seven gives the slope function of the curve in
terms of both 𝑥 and 𝑦. d𝑦 by d𝑥 is equal to four minus eight 𝑥𝑦 squared over
eight 𝑥 squared 𝑦 minus seven.
Next, we need to evaluate the slope
function at the point negative one, one. Substituting 𝑥 equals negative one
and 𝑦 equals positive one gives d𝑦 by d𝑥 equals four minus eight multiplied by
negative one multiplied by one squared over eight multiplied by negative one squared
multiplied by one minus seven. That simplifies to four plus eight
over eight minus seven, which is 12 over one or simply 12. We now know that the slope of the
tangent is 12, and it passes through the point with coordinates negative one,
one.
We can therefore find the equation
of the tangent using the point–slope form of the equation of a straight line. That’s 𝑦 minus 𝑦 one equals 𝑚
multiplied by 𝑥 minus 𝑥 one. Substituting 12 for the slope 𝑚
and negative one, one for 𝑥 one, 𝑦 one gives 𝑦 minus one equals 12 multiplied by
𝑥 minus negative one. Distributing the 12 over the
parentheses on the right-hand side gives 𝑦 minus one equals 12𝑥 plus 12. And then collecting all terms on
the left-hand side of the equation by subtracting 12𝑥 and 12 from each side gives
𝑦 minus 12𝑥 minus 13 equals zero.
So, by using implicit
differentiation to find the slope function of this curve and then evaluating the
slope function at the given point, we found that the equation of the tangent to the
curve four 𝑥 squared 𝑦 squared minus four 𝑥 minus seven 𝑦 minus one equals zero
at the point negative one, one is 𝑦 minus 12𝑥 minus 13 equals zero.
