Question

Find the derivative of the function.
f(t)=t cos (t)+t²sin (t)
fʹ(t)=

Answer 1

The derivative of the function f(t) = t cos(t) + t²sin(t) can be found by applying the product rule to each term, leading to the simplified final derivative: f′(t) = (1+t²) cos(t) + t sin(t).

Step-by-step explanation:

To find the derivative of the function f(t) = t cos(t) + t²sin(t), you apply the product rule and the chain rule of differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule is used to differentiate composite functions.

Applying the product rule to the first term t cos(t), we get:

• Derivative of t times cos(t) which is 1 × cos(t)
• Plus t times the derivative of cos(t) which is t × (-sin(t))

For the second term t²sin(t), we apply the product rule again:

• Derivative of t² times sin(t) which is 2t × sin(t)
• Plus t² times the derivative of sin(t) which is t² × cos(t)

Combining these, we get the final derivative:

f′(t) = cos(t) - t sin(t) + 2t sin(t) + t² cos(t)

Simplifying, the derivative of the function f(t) = t cos(t) + t²sin(t) is:

f′(t) = (1+t²) cos(t) + (2t - t) sin(t)

Which further simplifies to:

f′(t) = (1+t²) cos(t) + t sin(t)