In Exercises 59–86, find the derivative of y with respect to t...

Question

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

69. y = 2^(sin 3t)

Accepted Answer

Welcome back, everyone. Find D Y divided by DX4 Y equals 5 the power of sin of 2 X. For this problem we want to find D Y divided by D X, which is the derivative of Y with respect to X. We have to recognize that 5 to the power of sine of 2 X is a composite function. First of all, we have a form of 8 to the power of X, then we have sign of A X, and then we have a function 2 X, right? So we have the outer function, the middle function, and the inner function. So we want to apply the chain rule to differentiate it. According to the chain rule, we first will differentiate the outer function. 5 to the power of sign of two Xs. With respect to the middle function which is sin of 2 X. We then multiply by the derivative of the middle function so that be the derivative of sine of 2 X. With respect to the inner function. So with respect to 2 X. And finally, we multiply by the derivative of the inner function with respect to X. So we want to find these 3 derivatives. Let's begin with the derivative of 5 to the power of sin of 2 X. With respect to sign of two acts. And this is a simple exponential function. Let's remember it's derivative. We take that function itself, so that would be 5 to the power of sin of 2 X. Multiplied by LN of the base, so LN of 5, right? And this is our first derivative that we want to use. The next derivative in the chain rule is the derivative of sine of 2 X. With respect to the angle to X. Remember that the derivative of sine is cosine, and here we are getting cosine of 2 X. And finally, the derivative of 2 X with respect to X. We can find that by factoring out 2 and D X divided by DX is 1, so we basically get 2. Well done, so now we can multiply these to show that D Y divided by DX is equal to 2 multiplied by LN of 5, we can combine the constants. Multiplied by. Cosine Of two acts. Multiplied by the remaining term, which is 5 to the power of sine of 2 X. And that's our final derivative for this problem. Thank you for watching.

In Exercises 59–86, find the derivative of y with respect to the given independent variable.69. y = 2^(sin 3t)

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In Exercises 59–86, find the derivative of y with respect to the given independent variable.
69. y = 2^(sin 3t)