Differentiate the function: $f\left(t\right)={ln\left(t\right)}^{2}cos\left(t\right)$. Which of the following is the correct derivative $f^{\prime }\left(t\right)$?

• A. $2\frac{ln\left(t\right)}{t}cos\left(t\right)-{ln\left(t\right)}^{2}sin\left(t\right)$
• B. $2\frac{ln\left(t\right)}{t}cos\left(t\right)+{ln\left(t\right)}^{2}sin\left(t\right)$
• C. $2\frac{ln\left(t\right)}{t}sin\left(t\right)+{ln\left(t\right)}^{2}cos\left(t\right)$
• D. $2ln\left(t\right)cos\left(t\right)+{ln\left(t\right)}^{2}sin\left(t\right)$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the function f(t) = (ln(t))^2 * cos(t) is a product of two functions: u(t) = (ln(t))^2 and v(t) = cos(t). To differentiate this, we will use the product rule: (uv)' = u'v + uv'.
2. Step 2: Differentiate u(t) = (ln(t))^2. Use the chain rule: the derivative of (ln(t))^2 is 2 * ln(t) * (d/dt(ln(t))). Since d/dt(ln(t)) = 1/t, we get u'(t) = 2 * ln(t) / t.
3. Step 3: Differentiate v(t) = cos(t). The derivative of cos(t) is v'(t) = -sin(t).
4. Step 4: Apply the product rule: f'(t) = u'(t) * v(t) + u(t) * v'(t). Substitute u'(t), u(t), v(t), and v'(t) into the formula. This gives f'(t) = (2 * ln(t) / t) * cos(t) + (ln(t))^2 * (-sin(t)).
5. Step 5: Simplify the expression: f'(t) = 2 * ln(t) / t * cos(t) - (ln(t))^2 * sin(t). Compare this result with the given options to identify the correct answer.