find the derivatives, dy/dx, for the following functions :
y=(1+sin^2{2 exp (3x)})^5y
please help me to solve this assignment.
thank's
Are you sure the right hand of the equation ends with y? If so, you do not have an explicit y(x) function.
It can be differentiated implicitly, but is rather a mess.
yes i'm sure..the assignment like that.
wow. that's a nasty one.
Recall that, extending the power and exponent rules, if u and v are functions of x, then if
y = u^v
dy/dx = v*u^(v-1) du/dx + ln(u) * u^v * dv/dx
So, letting
u = 1+sin^2(2e^3x)
y = u^5y
y' = 5y * u^(5y-1) u' + ln(u) * u^5y * 5y'
Now, u' = 2sin(2e^3x)cos(2e^3x)*(6e^3x)
and I'm sure you can take it from there. Good luck!
thank's steve :)
To find the derivative, dy/dx, for the given function y=(1+sin^2{2 exp (3x)})^5y, we will use the chain rule.
Step 1: Identify the basic functions and their derivatives:
Let's start by breaking down the function into its components:
1. The outer function is (1 + sin^2{2 exp (3x)})^5y.
2. The inner function is 1 + sin^2{2 exp (3x)}.
Now, we can find the derivative of each part individually.
Derivative of the outer function:
The derivative of (1 + sin^2{2 exp (3x)})^5y with respect to y is just 1 since y is treated as a constant.
So, the derivative of the outer function is 1.
Derivative of the inner function:
To find the derivative of 1 + sin^2{2 exp (3x)}, we need to apply the chain rule.
Let u = 2 exp (3x).
So, the inner function can be rewritten as 1 + sin^2(u).
Derivative of sin^2(u):
To find the derivative of sin^2(u), we can use the chain rule.
Let's define v = sin(u).
Now, sin^2(u) can be written as v^2.
The derivative of v^2 is 2v.
Since v = sin(u), we can substitute v back in to get 2sin(u).
Derivative of u:
To find the derivative of u with respect to x, we need to apply the chain rule.
Let's define w = 2 exp (3x), so u = w.
The derivative of w with respect to x is given by the exponential function's derivative, which is e^w multiplied by the derivative of w with respect to x.
So, the derivative of w with respect to x is 3e^(2 exp(3x)).
Now, let's substitute back in to find the derivative of u with respect to x: du/dx = (dw/dx) = 3e^(2 exp(3x)).
Finally, substituting all the derivatives back into the chain rule formula, we have:
dy/dx = (d(1 + sin^2(u))/du) * (du/dx) * (d(1 + sin^2{2 exp (3x)})^5y/dy)
= 2sin(u) * 3e^(2 exp(3x)) * (1)
= 6sin(u)e^(2 exp(3x))
Now, we just need to substitute back in the expressions for u and y to simplify the final answer.