Video Transcript
If 𝑦 is equal to negative two times 𝑥 to the fourth power plus six times the sin of 𝑥 over two plus the cos of 𝜋 by four, find the derivative of 𝑦 with respect to 𝑥.
We need to find the derivative of 𝑦 with respect to 𝑥, and we can see that 𝑦 is given as the sum of three terms. It’s a polynomial term plus a trigonometric term plus another trigonometric term evaluated at a value of 𝑥. Since we know how to differentiate each of these three terms, we can find this by just differentiating each term separately. To start, remember, d𝑦 by d𝑥 is the derivative of 𝑦 with respect to 𝑥. In other words, this is equal to the derivative of negative two 𝑥 to the fourth power plus six times the sin of 𝑥 over two plus the cos of 𝜋 by four with respect to 𝑥.
And now, instead of evaluating this derivative as a whole, we’ll evaluate this derivative term by term. Let’s start with our first derivative of negative two 𝑥 to the fourth power with respect to 𝑥. This is the derivative of a polynomial, so we can do this by using the power rule for differentiation. We recall this tells us, for any real constants 𝑎 and 𝑛, the derivative of 𝑎𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑎 times 𝑛 times 𝑥 to the power of 𝑛 minus one. We multiply by our exponent of 𝑥 and then reduce this exponent by one.
In our case, the exponent of 𝑥 is equal to four. So we need to multiply by this exponent of four and then reduce this exponent by one. This gives us negative two times four multiplied by 𝑥 to the power of four minus one. And of course, we can simplify this entire expression to negative eight 𝑥 cubed. To evaluate the derivative of six times the sin of 𝑥 over two with respect to 𝑥, we need to notice one small piece of manipulation. We can rewrite 𝑥 over two as one-half multiplied by 𝑥.
And in doing this, we can now see that this derivative is a standard trigonometric derivative result. We know for any real constants 𝑎 and 𝐾, the derivative of 𝑎 times the sin of 𝐾𝑥 with respect to 𝑥 is equal to 𝑎𝐾 times the cos of 𝐾𝑥. In our case, we can see that the coefficient of 𝑥 is equal to one-half, so we’ll use this result with 𝐾 set to be equal to one-half. So by using our value of 𝑎 equal to six and 𝐾 equal to one-half, we get six times one-half multiplied by the cos of one-half 𝑥. And then, of course, we can simplify and rewrite this as three times the cos of 𝑥 over two.
Now, all we have to do is evaluate the derivative of the cos of 𝜋 by four with respect to 𝑥. But the cos of 𝜋 by four is equal to a constant. In fact, we know it’s equal to the square root of two divided by two. And we know that constants don’t vary as our value of 𝑥 varies. In other words, the rate of change of this constant with respect to 𝑥 is equal to zero, so its derivative with respect to 𝑥 will also be equal to zero. And because this is equal to zero, we don’t need to include this. Therefore, given that 𝑦 is equal to negative two 𝑥 to the fourth power plus six times the sin of 𝑥 over two plus the cos of 𝜋 by four, we were able to show that d𝑦 by d𝑥 is equal to negative eight 𝑥 cubed plus three times the cos of 𝑥 over two.
