Video: Finding the Equation of the Tangent to the Curve of a Function Defined Implicitly Given the Slope of the Tangent

Find the equation of the tangent to 9𝑦² = βˆ’ 7π‘₯ + 9 that has slope 7/18.

04:35

Video Transcript

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

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

So we’re 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 π‘Žπ‘₯ is just π‘Ž, 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 we’re differentiating this function of 𝑦, with respect to π‘₯. So, we have to apply the chain rule.

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

So, we have the 18𝑦 multiplied by d𝑦 by dπ‘₯ equals negative seven. So now, we have an equation involving d𝑦 by dπ‘₯. So, let’s rearrange to make d𝑦 by dπ‘₯ the subject. We divide both sides by 18𝑦. And we get d𝑦 by dπ‘₯ equals negative seven over 18𝑦.

Now remember, in the question we were told that the slope d𝑦 by dπ‘₯ is equal to seven over 18. So, replacing d𝑦 by dπ‘₯ by seven over 18, we have that seven over 18 equals negative seven over 18𝑦. And, solving for 𝑦 gives us that 𝑦 equals negative one. Substituting this back into the original equation, nine 𝑦 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, 𝑦 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. 𝑦 nought equal to negative one. And the slope π‘š equal to seven over 18. We have that the tangent line is 𝑦 minus negative one equals seven over 18 multiplied by π‘₯ minus zero. 𝑦 minus negative one is just 𝑦 plus one. And, expanding the brackets on the right-hand side gives us that 𝑦 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 18𝑦 minus seven π‘₯ plus 18 equals zero. And, that gives us our final answer.

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