Final answer:
To find the derivative of the given function r=sin(θ⁸) cos (2 θ), we can use the chain rule. The derivative is given by cos(θ⁸) cos(2θ) multiplied by 8θ⁷.
Step-by-step explanation:
To find the derivative of the given function, we can use the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of this composition is given by the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.
In this case, the outer function is r =
sin(θ&sup8;) cos (2 θ). And the inner function is θ&sup8;.
To find the derivative, we first differentiate the outer function with respect to the inner function, which gives us:
d(r)/d(θ) = cos(θ&sup8;) cos(2θ) * d(θ&sup8;)/d(θ)
We then differentiate the inner function, θ&sup8;, with respect to θ, which gives us:
d(θ&sup8;)/d(θ) = 8θ&sup7;
Finally, we substitute this result back into the previous equation to find the derivative:
d(r)/d(θ) = cos(θ&sup8;) cos(2θ) * 8θ&sup7;