y 6x5xSolution ddx6 x5x Use the chain rule ddx6 x5x duv d

Solution

d/dx((6 x+5)^x) | Use the chain rule, d/dx((6 x+5)^x) = ( du^v)/( du) ( du)/( dx)+( du^v)/( dv) ( dv)/( dx), where u = 6 x+5, v = x and ( du^v)/( du) = u^(-1+v) v, ( du^v)/( dv) = u^v log(u): = | x (6 x+5)^(x-1) (d/dx(6 x+5))+(6 x+5)^x log(6 x+5) (d/dx(x)) | Differentiate the sum term by term and factor out constants: = | x (6 x+5)^(x-1) (6 (d/dx(x))+d/dx(5))+(6 x+5)^x log(6 x+5) (d/dx(x)) | The derivative of 5 is zero: = | x (6 x+5)^(x-1) (6 (d/dx(x))+0)+(6 x+5)^x log(6 x+5) (d/dx(x)) | The derivative of x is 1: = | (6 x+5)^x log(6 x+5) (d/dx(x))+6 x (6 x+5)^(x-1) | The derivative of x is 1: = | 6 x (6 x+5)^(x-1)+(6 x+5)^x log(6 x+5)