differentiate the function. f(x)=ln(144 sin²(x)) f(x)= resources read it submit answer 4. -/1 points…

Explanation:

Step1: Differentiate outer - function for (f(x)=\ln(144\sin^{2}(x)))

Let (u = 144\sin^{2}(x)). The derivative of (y=\ln(u)) with respect to (u) is (\frac{1}{u}).

Step2: Differentiate inner - function

The derivative of (u = 144\sin^{2}(x)) with respect to (x) using the chain - rule. First, if (v=\sin(x)), then (u = 144v^{2}). The derivative of (u) with respect to (v) is (288v), and the derivative of (v=\sin(x)) with respect to (x) is (\cos(x)). So, (\frac{du}{dx}=288\sin(x)\cos(x)).

Step3: Apply chain - rule

By the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Substituting (u = 144\sin^{2}(x)) and (\frac{du}{dx}=288\sin(x)\cos(x)) into the chain - rule formula, we get (f^{\prime}(x)=\frac{288\sin(x)\cos(x)}{144\sin^{2}(x)}=\frac{2\cos(x)}{\sin(x)} = 2\cot(x)).

Step4: Differentiate (y=\frac{11}{\ln(x)}) using quotient - rule

The quotient - rule states that if (y=\frac{g(x)}{h(x)}), then (y^{\prime}=\frac{g^{\prime}(x)h(x)-g(x)h^{\prime}(x)}{h^{2}(x)}). Here, (g(x) = 11), so (g^{\prime}(x)=0), and (h(x)=\ln(x)), so (h^{\prime}(x)=\frac{1}{x}).

Step5: Calculate the derivative

(y^{\prime}=\frac{0\times\ln(x)-11\times\frac{1}{x}}{\ln^{2}(x)}=-\frac{11}{x\ln^{2}(x)})

Answer:

(f^{\prime}(x)=2\cot(x)) (y^{\prime}=-\frac{11}{x\ln^{2}(x)})