Understanding the Function
The function given is y = e^x + e^(-x). This is a common hyperbolic function, specifically, y = 2cosh(x).
Finding the Derivative
To find dy/dx, we differentiate y with respect to x:
- The derivative of e^x is e^x.
- The derivative of e^(-x) is -e^(-x).
Thus,
- dy/dx = e^x - e^(-x).
Calculating y^2
Next, we compute y^2:
- y^2 = (e^x + e^(-x))^2 = e^(2x) + 2 + e^(-2x).
Finding Root of y^2
We need to find the square root of y^2:
- √(y^2) = √(e^(2x) + 2 + e^(-2x)) = e^x + e^(-x) = y.
Substituting in the Expression
Now, we substitute dy/dx and √(y^2) into the expression:
- dy/dx - √(y^2) - 4 = (e^x - e^(-x)) - (e^x + e^(-x)) - 4.
Simplifying the Expression
This simplifies to:
- dy/dx - y - 4 = e^x - e^(-x) - e^x - e^(-x) - 4.
- This results in -2e^(-x) - 4.
Final Result
Thus,
- dy/dx - √(y^2) - 4 = -2e^(-x) - 4.
This provides a complete understanding of the differentiation process and the manipulation of the function.