Video Transcript
If 𝑦 is equal to negative two times the sec of two 𝑥, determine the rate of change of 𝑦 when 𝑥 is equal to 11𝜋 by six.
We’re given 𝑦 as a function in 𝑥, and we’re asked to determine the rate of change of 𝑦 when 𝑥 is equal to 11𝜋 by six. First, recall the rate of change of a function is just its derivative with respect to its variable. In this case, 𝑦 is a function in 𝑥. So, we want to calculate the rate of change of 𝑦 with respect to 𝑥. This is d𝑦 by d𝑥. However, we want to do this specifically when 𝑥 is equal to 11𝜋 by six. So, we need to find an expression for d𝑦 by d𝑥 and then substitute 𝑥 is equal to 11𝜋 by six into this expression.
So, d𝑦 by d𝑥 will be the derivative of 𝑦 with respect to 𝑥. That’s the derivative of negative two sec of two 𝑥 with respect to 𝑥. And there’s a few different ways we could differentiate this. For example, we could use our trigonometric identities to rewrite this expression as negative two divided by the cos of two 𝑥. And then, we could differentiate this expression by using the quotient rule, the general power rule, or the chain rule, and this would work. However, we’ve already done this in the general case to derive the following result.
We know for any real constant 𝑎, the derivative of the sec of 𝑎𝑥 with respect to 𝑥 is equal to 𝑎 times the tan of 𝑎𝑥 multiplied by the sec of 𝑎𝑥. First, we can see our derivative is not quite in this form. We have a constant factor of negative two. However, we can just take this outside of our derivative. So, we can rewrite this as negative two times the derivative of the sec of two 𝑥 with respect to 𝑥. And now, we can just apply our rule with the value of 𝑎 equal to two. So, by setting the value of 𝑎 equal to two, and remember, we have a coefficient of negative two, we get negative two times two tan of two 𝑥 multiplied by the sec of two 𝑥. And of course, we can simplify this. Negative two multiplied by two is equal to negative Four. So, we were able to show d𝑦 by d𝑥 is equal to negative four tan of two 𝑥 times the sec of two 𝑥.
But remember, the question is asking us to find the rate of change of 𝑦 when 𝑥 is equal to 11𝜋 by six. So, we need to substitute 𝑥 is equal to 11𝜋 by six into this expression. Substituting 𝑥 is equal to 11𝜋 by six into our expression for d𝑦 by d𝑥, we get d𝑦 by d𝑥 11𝜋 by six is equal to negative four times the tan of two multiplied by 11𝜋 by six multiplied by the sec of two times 11𝜋 by six. And of course, we can simplify this. First, we know that two times 11𝜋 by six is just equal to 11𝜋 by three. This gives us negative four times the tan of 11𝜋 by three multiplied by the sec of 11𝜋 by three. We could just evaluate this expression using a calculator; however, it’s not necessary.
First, recall the tangent function is periodic about 𝜋. This means we can add and subtract multiples of 𝜋 from our argument, and it won’t change the value of our expression. So, to make this expression easier to calculate, we’ll subtract three 𝜋 from the argument of the tangent function. And we can do something very similar for the secant function. However, the secant function is one divided by the cosine. So, it’s periodic about two 𝜋. So, we could only add and subtract multiples of two 𝜋. So, instead, we’ll subtract four 𝜋 from our argument of the secant function. We know 11𝜋 by three minus three 𝜋 is two 𝜋 by three and 11𝜋 by three minus four 𝜋 is negative 𝜋 by three. So, this gives us negative four tan of two 𝜋 by three multiplied by the sec of negative 𝜋 by three.
And now, we’re almost ready to evaluate this expression. We’ll use the following trigonometric identity. We know the sec of 𝜃 is equivalent to one divided by the cos of 𝜃. So, instead of multiplying by the sec of negative 𝜋 by three, we can instead divide by the cos of negative 𝜋 by three. This gives us negative four tan of two 𝜋 by three all divided by the cos of negative 𝜋 by three. And now, this is entirely written in terms of fundamental angles. So, we should know these values. The tan of two 𝜋 by three is negative root three, and the cos of negative 𝜋 by three is one-half. So, this simplifies to give us negative four times negative root three over one-half. And we can simplify this. First, dividing by one-half is the same as multiplying by two. Then, in our numerator, negative eight multiplied by negative root three is just eight root three, which is our final answer.
Therefore, we were able to show if 𝑦 is equal to negative two sec of two 𝑥, then the rate of change of 𝑦 when 𝑥 is equal to 11𝜋 by six is eight root three.
