What is the derivative of the expression (10+lgx^10+e^10x)^10?
We have to find the derivative of y = (10 + lg (x^10) + e^10x)^10.
We use the chain rule to find the derivative of y.
y' = 10 * (10 + lg (x^10) + e^10x)^9 * (10 / x + 10*e^10x)
=> 10 * (10 + lg (x^10) + e^10x)^9 * 10 (1/ x + e^10x)
=> 100*(10 + lg (x^10) + e^10x)^9*(1/ x + e^10x)
The required derivative is 100*(10 + lg (x^10) + e^10x)^9*(1/x + e^10x)
Since the given expression represents a composed function, we'll evaluate its derivative applying chain rule.
We'll put 10+lgx^10+e^10x = t
y = t^10
We'll differentiate y with respect to t:
dy/dt = d(t^10)/dt
dy/dt = 10t^9
We'll differentiate t with respect to x:
dt/dx = d(10+lgx^10+e^10x)/dx
dt/dx = 10/x + 10*e^10x
dy/dx = 10t^9*(10/x + 10*e^10x)
We'll substitute back t:
dy/dx = 10*[(10+lgx^10+e^10x)^9]*(10/x + 10*e^10x)