Derivative of tan⁻¹(2-5x/5+2x)
Understanding the given function is crucial in finding its derivative. The function is tan⁻¹(2-5x/5+2x), where tan⁻¹ represents the inverse tangent function. To find the derivative of this function, we need to apply the chain rule and the derivative of the inverse tangent function.

Chain Rule
The chain rule states that if we have a composite function, f(g(x)), then the derivative is f'(g(x)) * g'(x). In this case, our composite function is tan⁻¹(2-5x/5+2x).

Derivative of Inverse Tangent Function
The derivative of tan⁻¹(u) with respect to u is 1/(1+u^2). Therefore, the derivative of tan⁻¹(2-5x/5+2x) with respect to (2-5x/5+2x) is 1/(1+(2-5x/5+2x)^2).

Putting It All Together
Now, applying the chain rule, the derivative of tan⁻¹(2-5x/5+2x) with respect to x is:
(1/(1+(2-5x/5+2x)^2)) * (d/dx(2-5x/5+2x))
Simplify the expression by finding the derivative of 2-5x/5+2x with respect to x, and substitute it back into the chain rule formula.
After simplifying the expression and performing the necessary calculations, you will have the derivative of tan⁻¹(2-5x/5+2x) with respect to x.