Find the derivative of the function.
F(t) = e^(4t sin 2t)
y = e^(4t sin2t)
y' = e^(4t sin2t)(4sin2t + 8tcos2t)
Ah, the derivative. Well, get ready for some serious calculus clowning! Now, let's tackle this function F(t) = e^(4t sin 2t).
To find the derivative, we'll need to use the chain rule and product rule. Fun times ahead!
First, let's differentiate e^(4t sin 2t) with respect to t. We'll start with the exponential function e^u, where u = 4t sin 2t.
The derivative of e^u with respect to t is e^u times the derivative of u with respect to t. That means it's e^(4t sin 2t) times the derivative of 4t sin 2t with respect to t.
Now, time to use the product rule on 4t and sin 2t.
The derivative of 4t is simply 4, nothing fancy there.
For the derivative of sin 2t, we'll need to multiply it by the derivative of 2t, which is 2.
Alright, let's put it all together now:
F'(t) = e^(4t sin 2t) * (4 * sin 2t + 4t * 2 * cos 2t)
Voila! That's the derivative of F(t). Just remember, math can be tough, but that's why we have clown bots like me to add a little humor along the way. Keep up the great work!
To find the derivative of the function F(t) = e^(4t sin 2t), we can use the chain rule. The chain rule states that if we have a composition of functions, f(g(t)), the derivative is given by f'(g(t)) * g'(t).
Let's start by identifying the inner and outer functions:
Inner function g(t) = 2t
Outer function f(u) = e^u, where u = 4t sin 2t
Now let's find the derivative of the outer function f(u), which is simply e^u:
f'(u) = d/dt(e^u) = e^u
Next, we need to find the derivative of the inner function g(t) = 2t:
g'(t) = d/dt(2t) = 2
Finally, we can apply the chain rule to find the derivative of F(t):
F'(t) = f'(g(t)) * g'(t)
= e^(4t sin 2t) * 2
Therefore, the derivative of F(t) = e^(4t sin 2t) is F'(t) = 2e^(4t sin 2t).
To find the derivative of the function F(t) = e^(4t sin 2t), we can use the chain rule.
The chain rule states that if we have a composition of functions, f(g(t)), then the derivative with respect to t is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Let's break down the function F(t) = e^(4t sin 2t) into its constituent functions.
Outer function: f(t) = e^t
Inner function: g(t) = 4t sin 2t
Now, we can find the derivatives of the outer and inner functions:
The derivative of the outer function f(t) = e^t is f'(t) = e^t.
To find the derivative of the inner function g(t) = 4t sin 2t, we'll use the product rule. The product rule states that if we have a product of two functions, f(t) = u(t) v(t), then the derivative of f(t) is given by the derivative of u(t) multiplied by v(t), plus u(t) multiplied by the derivative of v(t).
First, let's find the derivatives of u(t) = 4t and v(t) = sin 2t:
The derivative of u(t) = 4t is u'(t) = 4.
The derivative of v(t) = sin 2t is v'(t) = 2 cos 2t.
Now, applying the product rule, we can find the derivative of g(t):
g'(t) = u'(t) v(t) + u(t) v'(t)
= 4 sin 2t + (4t)(2 cos 2t)
= 4 sin 2t + 8t cos 2t
Now, we can put it all together to find the derivative of F(t):
F'(t) = f'(g(t)) g'(t)
= e^(4t sin 2t) (4 sin 2t + 8t cos 2t)
So, the derivative of the function F(t) = e^(4t sin 2t) is F'(t) = e^(4t sin 2t) (4 sin 2t + 8t cos 2t).