Question

find the derivative of the function.
y = cot²(sin(θ))
y=

resources
read it watch it

submit answer

1. - / 1 points

details my notes ask your teacher practice ano

find the derivative of the function.
f(t) = sin²(e^(sin²(t)))
f(t)=

Explanation:

Step1: Apply chain - rule for outer function

Let \(u = \sin(\theta)\), then \(y=\cot^{2}(u)\). The derivative of \(y\) with respect to \(u\) using the power - rule \((x^{n})'=nx^{n - 1}\) and the derivative of \(\cot(u)\) is \(-\csc^{2}(u)\). So \(\frac{dy}{du}=2\cot(u)\times(-\csc^{2}(u))=- 2\cot(u)\csc^{2}(u)\).

Step2: Differentiate inner function

The derivative of \(u = \sin(\theta)\) with respect to \(\theta\) is \(\frac{du}{d\theta}=\cos(\theta)\).

Step3: Apply chain - rule

By the chain - rule \(\frac{dy}{d\theta}=\frac{dy}{du}\times\frac{du}{d\theta}\). Substituting \(\frac{dy}{du}=-2\cot(u)\csc^{2}(u)\) and \(u = \sin(\theta)\) and \(\frac{du}{d\theta}=\cos(\theta)\), we get \(y'=-2\cot(\sin(\theta))\csc^{2}(\sin(\theta))\cos(\theta)\).

Answer:

\(-2\cot(\sin(\theta))\csc^{2}(\sin(\theta))\cos(\theta)\)