Find the derivative of the function

1)f(t)=tan(e^2t)+e^tan2t
f'(t)=

[im not quite sure with this. im assuming the implicit differentiation]

2) F(t)=e^(8tsin2t)

[for this one i got 2e^(8t)(4sin(2t)+cos(2t)) but it is incorrect.

1) f'(t)=2*e^2t*(sec(e^2t))^2 + 2*e^tan2t*(sec(2t))^2

2)F'(t)=e^(8tsin2t)*(8sin2t+18tcos(2t))

#1: no impliocit, just use the chain rule

f=tan(u) + e^v
f' = sec^2(u) u' + e^v v'
= sec^2(e^2t) * 2e^2t + e^(tan2t) * sec^2(2t) * 2

#2: f = e^u
f' = u^u u'
= e^8tsin2t * (8sin2t + 16tcos2t)

To find the derivative of the given functions, we can use the chain rule and product rule. Let's take a look at each function separately:

1) f(t) = tan(e^2t) + e^tan2t

To find the derivative of this function, we'll use the chain rule. The derivative of the tangent function is sec^2(x), and the derivative of e^x is e^x:

f'(t) = sec^2(e^2t) * d/dt(e^2t) + e^(tan2t) * d/dt(tan2t)

= sec^2(e^2t) * 2e^2t + e^(tan2t) * sec^2(2t) * d(tan2t)/dt

Now, let's find the derivative of tan(2t) using the chain rule. The derivative of tan(x) is sec^2(x):

d(tan2t)/dt = 2sec^2(2t)

Plugging this back into the equation, we have:

f'(t) = sec^2(e^2t) * 2e^2t + e^(tan2t) * sec^2(2t) * 2sec^2(2t)

Simplifying further, we get:

f'(t) = 2e^2t * sec^2(e^2t) + 2e^(tan2t) * sec^4(2t)

2) F(t) = e^(8tsin2t)

To find the derivative of this function, we can use the chain rule. The derivative of e^x is e^x, and the derivative of 8tsin(2t) is 8sin(2t) + 16t*cos(2t):

F'(t) = e^(8tsin2t) * d/dt(8tsin2t)

= e^(8tsin2t) * (8sin(2t) + 16t*cos(2t))

Therefore, the derivative of F(t) is:

F'(t) = e^(8tsin2t) * (8sin(2t) + 16t*cos(2t))

I hope this helps! Let me know if you have any further questions.

To find the derivative of a function, we can use various differentiation rules such as the chain rule, product rule, and power rule. Let's go through each of the given functions and find their derivatives.

1) f(t) = tan(e^2t) + e^tan(2t)

To find the derivative of f(t), we can use the chain rule. First, let's differentiate each term separately and then add them together.

The derivative of the first term, tan(e^2t), can be calculated using the chain rule. We start by differentiating the outer function, tan(x), with respect to the inner function, e^2t. The derivative of tan(x) is sec^2(x). Multiplying this by the derivative of the inner function, we get sec^2(e^2t) * (2e^2t).

For the second term, e^tan(2t), we need to use the chain rule again. The outer function is e^x, whose derivative is simply e^x. However, we need to multiply this by the derivative of the inner function, tan(2t). The derivative of tan(x) is sec^2(x), so the derivative of tan(2t) is sec^2(2t) * 2.

Adding the derivatives, we get:

f'(t) = sec^2(e^2t) * (2e^2t) + e^tan(2t) * sec^2(2t) * 2

2) F(t) = e^(8tsin(2t))

To find the derivative of F(t), we need to use the chain rule again. The outer function is e^x, whose derivative is simply e^x. However, we need to multiply this by the derivative of the inner function, 8tsin(2t).

Differentiating 8tsin(2t) with respect to t, we can start by using the product rule. We differentiate 8t (which is a simple power function) to get 8, and leave sin(2t) unchanged. Then we differentiate sin(2t) using the chain rule, resulting in 2cos(2t).

Multiplying all the terms together, we get:

F'(t) = e^(8tsin(2t)) * (8 + 2cos(2t))

Therefore, the correct derivative of F(t) is e^(8tsin(2t)) * (8 + 2cos(2t)).

I hope this helps! Let me know if you have any further questions.