What is the derivative of the expression (10+lgx^10+e^10x)^10?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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)

 

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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)

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