y = (x+√x+1)4 (x-1) what is the dy/dx=?  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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I will consider that 4 is the power of `(x + sqrt(x+1)), `  hence, differentiating with respect to x and using the product and chain rules yields:

`dy/dx = 4(x + sqrt(x+1))^(4-1)*(x + sqrt(x+1))'*(x - 1) + (x + sqrt(x+1))^4*(x-1)'`

`dy/dx = 4(x + sqrt(x+1))^3*(1 + 1/(2sqrt(x+1)))*(x - 1) + (x + sqrt(x+1))^4`

You need to factor out `(x + sqrt(x+1))^3`  such that:

`dy/dx = (x + sqrt(x+1))^3*((4x-4)*(1 + 1/(2sqrt(x+1))) + x + sqrt(x+1))`

`dy/dx = (x + sqrt(x+1))^3*(4x + (4x)/(2sqrt(x+1)) - 4 - 4/(2sqrt(x+1)) + x + sqrt(x+1))`

`dy/dx = (x + sqrt(x+1))^3*(5x + (2x)/(sqrt(x+1)) - 4 - 4/(2sqrt(x+1)) + sqrt(x+1))`

Hence, considering`(x + sqrt(x+1))^4, dy/dx = (x + sqrt(x+1))^3*(5x + (2x)/(sqrt(x+1)) - 4 - 4/(2sqrt(x+1)) + sqrt(x+1)).`

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