Calculate the derivative of the following functions.
y =...

Question

Calculate the derivative of the following functions.

y = (p+3)² sin p²

Accepted Answer

Welcome back everyone. In this problem, we want to find the derivative of the function Y equals X + 55 multiplied by the sin of X 5. A says it's 5 multiplied by X + 55 multiplied by the sine of X 5th, plus 5 X 4th multiplied by X + 5 to 5th, cosine x to the 5th. B says it's 5 multiplied by X + 5 to 4th. S X 5 plus X + 5 5th, cost X 5. C25 multiplied by X + 5 to 4th. Sign X to the 5th plus 5 X 4th multiplied by X + 5 to 5th. Cox 5. And the D says it's 5 multiplied by X + 5 to 4th, X 5th, plus 5 X 4th, multiplied by X + 5 to 5th, cost X to the 5th. Now, when we look at our function Y, we can tell that it is a product, and thus to differentiate, we can use the product rule of differentiation. Recall that if we have a function Y equals UV where U and V are functions of X, then the derivative of Y with respect to X is going to be equal to VDUD X. Plus UDVDX. In other words, we're going to differentiate both terms and multiply them by the opposite original term. So if we let you be equal to X + 5 raised to the 5th, OK. And we let V be equal to sin x to the 5th. Let's differentiate both of these terms and plug them into the product rule. Now let's start with U equals x + 5 to 5th. This is a composite function. So we can differentiate it using the chain rule. And to do that, we're basically going to differentiate the other function. So we're going to bring down the power of 5, write backward function X + 5, and subtract 1 from the power. So 5 minus 1 is 4, and then we're going to multiply it by the derivative of the inner function. The derivative of X + 5 is 1. So thus, that tells us DUD X equals 5 multiplied by X + 5 raised to the 4th. Next to differentiate V is a trigonometric function, OK, and the derivative of the sine function is equal to the cosine of that function multiplied by the derivative of the function inside. In other words, this is going to be equal to the cosine of x 5 multiplied by the derivative of X 5th, which by the poor rule would be equal to 5x the 4th. Thus DVD X equals 5 x 4 multiplied by the cosine of x 5th. Now that we have DUDX and DVDX we can multiply them by the opposite original terms to solve for DUIDX. Because no, this means that the UIDX. Is going to be equal to V5 X to the 5th. Multiplied by DUDX 5 multiplied by X + 5 to the 4th, OK. Plus U X + 55 multiplied by DVD X. 5 X to the 4th plus X 5th. And no, we can rewrite this as 5 multiplied by X + 5 to the 4th. Multiplied by the sign of x to the 5th. Plus 5 X 4th, multiplied by X plus 5 to 5th, multiplied by the cosine of X 5th. Now if we compare our result to our answer choices, this tells us that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Calculate the derivative of the following functions.
y = (p+3)² sin p²

Key Concepts

Derivative

Product Rule

Chain Rule

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