Video Transcript
Find the derivative of 𝑦 with respect to 𝑥, given that 𝑦 is equal to 𝑥 to the ninth power plus eight times the sin of five 𝑥.
We’re given that 𝑦 is the sum of two functions. It’s 𝑥 to the ninth power plus eight times the sin of five 𝑥. We need to use this to find the derivative of 𝑦 with respect to 𝑥. And in fact, we know how to do this term by term. First, we can differentiate polynomials by using the power rule for differentiation. Next, we know how to differentiate simple trigonometric functions. So, we can evaluate this derivative directly. First, d𝑦 by d𝑥 is equal to the derivative of 𝑥 to the ninth power plus eight times the sin of five 𝑥 with respect to 𝑥.
Next, we’ll differentiate this term by term. We need to find the derivative of 𝑥 to the ninth power with respect to 𝑥, and we need to add the derivative of eight sin five 𝑥 with respect to 𝑥. And now, we can evaluate each of these derivatives separately. First, to differentiate polynomials, we need to recall the power rule for differentiation. And this tells us for any real constants 𝑎 and 𝑛, the derivative of 𝑎 times 𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑎 times 𝑛 multiplied by 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 nine. So, we need to multiply by this exponent of nine and then reduce this exponent by one. This gives us nine 𝑥 to the eighth power.
We now need to evaluate the second derivative. To do this, we need to recall one of our standard trigonometric derivative results. For any real constants 𝑎 and 𝑘, the derivative of 𝑎 times the sin of 𝑘𝑥 with respect to 𝑥 is equal to 𝑎 times 𝑘 multiplied by the cos of 𝑘𝑥.
In our case, the coefficient of 𝑥 is equal to five. So, we use the value of 𝑘 is equal to five in our trigonometric derivative result. This gives us eight times five multiplied by the cos of five 𝑥. And of course, we can simplify this by evaluating eight times five to be equal to 40. This gives us nine 𝑥 to the eighth power plus 40 cos of five 𝑥.
Therefore, we were able to show if 𝑦 is equal to 𝑥 to the ninth power plus eight sin of five 𝑥, then d𝑦 by d𝑥 will be equal to nine 𝑥 to the eighth power plus 40 times the cos of five 𝑥.
