Question

differentiate the function. f(x)=ln(144 sin²(x)) f(x)= resources read it submit answer 4. -/1 points differentiate the function. y = 11/ln(x) y= resources read it

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:

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