Differentiate with respect to (t).
y = d cos(t) + (t^2)sin(t)
y' = -d sin(t) + 2t sin(t) + t^2 cos(t)
To differentiate the given function with respect to t, we need to apply the basic rules of differentiation. Here's a step-by-step approach:
Step 1: Identify the terms within the function
In the given function, we have two terms:
Term 1: d cos(t)
Term 2: (t^2)sin(t)
Step 2: Differentiate each term separately
For Term 1: d cos(t)
The derivative of cos(t) with respect to t is -sin(t). Since the coefficient 'd' is a constant, its derivative with respect to t is zero (d/dt (constant) = 0). Therefore, the derivative of Term 1 is -d sin(t).
For Term 2: (t^2)sin(t)
This term involves a product between (t^2) and sin(t). We need to apply the product rule of differentiation, which states that the derivative of a product is equal to the first term multiplied by the derivative of the second term, plus the second term multiplied by the derivative of the first term.
Let's break down the term:
First Term: (t^2)
Derivative of the first term: d/dt (t^2) = 2t
Second Term: sin(t)
Derivative of the second term: d/dt (sin(t)) = cos(t)
Now, applying the product rule:
Derivative of Term 2 = (First Term * Derivative of Second Term) + (Second Term * Derivative of First Term)
= (t^2 * cos(t)) + (sin(t) * 2t)
Step 3: Combine the derivatives of each term
To get the overall derivative of the function, we add the derivatives of the individual terms:
y' = -d sin(t) + (t^2 * cos(t)) + (sin(t) * 2t)
Therefore, the derivative of the given function with respect to t is:
y' = -d sin(t) + (t^2 * cos(t)) + (sin(t) * 2t)