To find the derivative of the function \(y = \sin^{2}(5x)\) with respect to \(x\), we will use the chain rule and the product rule of differentiation. Let's break this down step-by-step. ### Step 1: Identify the outer and inner functions In the function \(y = \sin^{2}(5x)\), we can identify: - The outer function is \(u^2\) where \(u = \sin(5x)\). - The inner function is \(u = \sin(5x)\). ### Step 2: Apply the chain rule The chain rule states that if you have a composite function \(y = f(g(x))\), then the derivative \(dy/dx\) is given by: \[ \frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \] In our case: - \(f(u) = u^2\) and its derivative \(f'(u) = 2u\). - \(g(x) = \sin(5x)\) and we need to find \(g'(x)\). ### Step 3: Differentiate the inner function To differentiate \(g(x) = \sin(5x)\), we again use the chain rule: - The derivative of \(\sin(v)\) is \(\cos(v)\), where \(v = 5x\). - The derivative of \(5x\) is \(5\). Thus, using the chain rule: \[ \frac{dg}{dx} = \cos(5x) \cdot 5 = 5\cos(5x) \] ### Step 4: Combine the derivatives Now we can combine the derivatives using the chain rule: \[ \frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = 2u \cdot \frac{dg}{dx} \] Substituting \(u = \sin(5x)\) and \(\frac{dg}{dx} = 5\cos(5x)\): \[ \frac{dy}{dx} = 2\sin(5x) \cdot (5\cos(5x)) = 10\sin(5x)\cos(5x) \] ### Final Answer Thus, the derivative of \(y = \sin^{2}(5x)\) with respect to \(x\) is: \[ \frac{dy}{dx} = 10\sin(5x)\cos(5x) \] ### Explanation of Options Now, let's evaluate the options provided: - **Option A: \(10 \sin(5x) \cos(5x)\)** - This is the correct answer as derived above. - **Option B: \(5 \sin(5x) \cos(5x)\)** - This is incorrect because it does not account for the factor of 2 from the outer function's derivative. - **Option C: \(2 \sin(5x) \cos(5x)\)** - This is incorrect because it also fails to include the factor of 5 from the derivative of the inner function. - **Option D: \(15 \sin(5x) \cos(5x)\)** - This is incorrect as it incorrectly combines the factors, leading to an incorrect coefficient. ### Common Pitfalls - **Forgetting the chain rule**: When dealing with composite functions, always remember to apply the chain rule correctly. - **Miscalculating derivatives**: Ensure that you differentiate each part of the function accurately, especially when using the chain rule. ### Revision Summary - Use the chain rule for composite functions: \( \frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \). - Differentiate the outer function and multiply by the derivative of the inner function. - Be careful with coefficients and ensure all parts of the function are accounted for. - Practice differentiating trigonometric functions and applying the chain rule to build confidence.