To find the derivative of the function \( f(t) = t^2 \sin(3t - 5) \) with respect to \( t \), we will use the **product rule** and the **chain rule** of differentiation. Let's break this down step-by-step. ### Step 1: Identify the components of the function The function \( f(t) \) is a product of two functions: - \( u(t) = t^2 \) - \( v(t) = \sin(3t - 5) \) ### Step 2: Apply the product rule The product rule states that if you have two functions \( u(t) \) and \( v(t) \), the derivative of their product is given by: \[ \frac{d}{dt}[u(t)v(t)] = u'(t)v(t) + u(t)v'(t) \] ### Step 3: Differentiate \( u(t) \) and \( v(t) \) 1. **Differentiate \( u(t) = t^2 \)**: \[ u'(t) = 2t \] 2. **Differentiate \( v(t) = \sin(3t - 5) \)**: To differentiate \( v(t) \), we need to use the chain rule. The chain rule states that if you have a composite function \( g(h(t)) \), then: \[ \frac{d}{dt}[g(h(t))] = g'(h(t)) \cdot h'(t) \] Here, \( g(x) = \sin(x) \) and \( h(t) = 3t - 5 \). - The derivative of \( g(x) = \sin(x) \) is \( g'(x) = \cos(x) \). - The derivative of \( h(t) = 3t - 5 \) is \( h'(t) = 3 \). Therefore, using the chain rule: \[ v'(t) = \cos(3t - 5) \cdot 3 = 3\cos(3t - 5) \] ### Step 4: Substitute back into the product rule formula Now we can substitute \( u(t) \), \( u'(t) \), \( v(t) \), and \( v'(t) \) back into the product rule: \[ f'(t) = u'(t)v(t) + u(t)v'(t) \] Substituting the values we found: \[ f'(t) = (2t)(\sin(3t - 5)) + (t^2)(3\cos(3t - 5)) \] ### Step 5: Simplify the expression Now we can simplify the expression: \[ f'(t) = 2t \sin(3t - 5) + 3t^2 \cos(3t - 5) \] ### Final Answer Thus, the derivative of \( t^2 \sin(3t - 5) \) with respect to \( t \) is: \[ f'(t) = 2t \sin(3t - 5) + 3t^2 \cos(3t - 5) \] ### Correct Option The correct option is **C**: \( 2t \sin(3t - 5) + 3t^2 \cos(3t - 5) \). ### Explanation of Other Options - **Option A: \( 6t \cos(3t - 5) \)**: This option incorrectly applies the product rule and does not account for the \( t^2 \) term correctly. It seems to suggest a derivative of a single function rather than a product. - **Option B: \( 2t \sin(3t - 5) - 3t^2 \cos(3t - 5) \)**: This option incorrectly uses subtraction instead of addition in the product rule application. The signs are wrong. - **Option D: \( 2t \sin(3t - 5) + t^2 \cos(3t) \)**: This option incorrectly differentiates the \( \cos \) term, failing to apply the chain rule correctly, and also has a missing factor of 3 in the cosine term. ### Revision Summary - Use the **product rule** for derivatives of products of functions. - Apply the **chain rule** for composite functions. - Carefully differentiate each component and combine results correctly. - Always check for simplification and correct application of rules to avoid common pitfalls.