Find the derivatives of the functions in Exercises 1–42.

Question

𝓻 = ( sin θ )²

( cos θ - 1 )

Accepted Answer

Hello. In this video, we are going to be calculating the derivative of the given function. The function given to us is G of Y. Equal to the quantity cosine of Y divided by sin of Y minus 3 race to the 8th power. Now, in order to take the derivative of this function, we are going to have to use the chain rule. Our outermost function is going to be this exponent of 8, and our innermost function is going to be cosine of Y divided by sin of Y minus 3. So in order to take the derivative by using the chain rule, We will go ahead and start by taking the derivative of the outermost function. In order to take the derivative of the outermost function, we will have to use the power rule. By the power rule, the derivative of the outermost function will be 8, multiplied by the original innermost function, which is cosine of Y divided by sin of Y minus 3. And this will be raised to the 7th power. Then what we need to do is we need to multiply this function by the derivative of the innermost function. Now, the derivative of the innermost function is going to need to use the quotient rule. By the quotient rule, the derivative of cosine of Y divided by sin Y minus 3 will be the quantity of Y minus 3 multiplied by the derivative of our numerator, which is going to be negative sign of Y. Then we will subtract the original numerator, which is cosine of Y, multiplied by the derivative of the denominator, which will also be cosine of Y. Then this will all be divided. By the original denominator squared. So sign of Y minus 3 quantity squared. Now, what we can do here is we can go ahead and move this product of negative sine of Y to the front of the numerator, and we can multiply the two cosine quantities together, and this quotient role will simplify to be negative sign of Y. Multiplied by the quantity. Sign of Y minus 3. Minus cosine square of y. Divided By sign of wine. -3 squared. So this is going to be the quotient rule of the innermost function from the chain rule. And so if we put this all together, the final version of our derivative will be G Y equal to 8, multiplied by the quantity, cosine of Y, divided by sin of Y minus 3, rates of the 7th power, multiplied by the entire quantity, negative sign of Y. Multiplied by sin of Y minus 3. Minus cosine squared of y. Divided By sign of Y minus 3 quantity squared. And this is going to be the derivative by using the chain rule. And with that being said, the overall solution is going to be C. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Find the derivatives of the functions in Exercises 1–42.

𝓻 = ( sin θ )²
( cos θ - 1 )

Key Concepts

Derivatives

Trigonometric Functions

Polar Coordinates

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