2nd derivative of
3t^2 Cos(4t)
Use the product rule for the first derivative, simplify that result, then do it again.
To find the second derivative of the function f(t) = 3t^2 * cos(4t), we will need to apply the chain rule and product rule. Let's go step by step.
Step 1: Find the first derivative of f(t):
To differentiate f(t), we need to apply the product rule, which states that d(uv)/dt = u*dv/dt + v*du/dt.
Using the product rule:
f'(t) = (3t^2)' * cos(4t) + (3t^2) * (cos(4t))'
Differentiating each term:
(3t^2)' = 6t
(cos(4t))' = -4 * sin(4t)
So, we have:
f'(t) = 6t * cos(4t) + 3t^2 * (-4 * sin(4t))
= 6t * cos(4t) - 12t^2 * sin(4t)
Step 2: Find the second derivative of f(t):
Now, to find the second derivative, we need to differentiate f'(t) using the product rule again.
Differentiating f'(t):
f''(t) = (6t * cos(4t) - 12t^2 * sin(4t))'
Applying the product rule:
= (6t)' * cos(4t) + 6t * (cos(4t))' - (12t^2)' * sin(4t) - 12t^2 * (sin(4t))'
Differentiating each term:
(6t)' = 6
(cos(4t))' = -4 * sin(4t)
(12t^2)' = 24t
(sin(4t))' = 4 * cos(4t)
Now, let's substitute these values back into the equation:
f''(t) = 6 * cos(4t) + 6t * (-4 * sin(4t)) - 24t * sin(4t) - 12t^2 * (4 * cos(4t))
Simplifying:
f''(t) = 6 * cos(4t) - 24t * sin(4t) - 24t * sin(4t) - 48t^2 * cos(4t)
Final answer:
The second derivative of f(t) = 3t^2 * cos(4t) is:
f''(t) = 6 * cos(4t) - 48t^2 * cos(4t) - 48t * sin(4t)