Question
Differentiate y = (x + 1)²/x
Answers
GPT 3.5
To differentiate the function y = (x + 1)²/x, we can begin by rewriting it as y = (x + 1)² * x^(-1).
Using the product rule, the derivative of y with respect to x will be:
dy/dx = [(x + 1)²]' * x^(-1) + (x + 1)² * (x^(-1))'.
To differentiate (x + 1)², we can use the chain rule. Let u = x + 1, so (x + 1)² = u². The derivative of u² with respect to x is 2u * du/dx.
Similarly, differentiating x^(-1) gives us (-1)x^(-2) using the power rule for differentiation.
Applying these results, we have:
dy/dx = [2(x + 1) * 1] * x^(-1) + (x + 1)² * (-1)x^(-2)
Simplifying further, we get:
dy/dx = 2(x + 1)/x + (-1)(x + 1)²/x²
Combining the terms, we have:
dy/dx = [2(x + 1) - (x + 1)²]/x²
Simplifying the expression inside the bracket, we get:
dy/dx = [2x + 2 - x² - 2x - 1]/x²
This simplifies to:
dy/dx = (-x² + 1)/x²
Therefore, the derivative of y = (x + 1)²/x is dy/dx = (-x² + 1)/x².
Using the product rule, the derivative of y with respect to x will be:
dy/dx = [(x + 1)²]' * x^(-1) + (x + 1)² * (x^(-1))'.
To differentiate (x + 1)², we can use the chain rule. Let u = x + 1, so (x + 1)² = u². The derivative of u² with respect to x is 2u * du/dx.
Similarly, differentiating x^(-1) gives us (-1)x^(-2) using the power rule for differentiation.
Applying these results, we have:
dy/dx = [2(x + 1) * 1] * x^(-1) + (x + 1)² * (-1)x^(-2)
Simplifying further, we get:
dy/dx = 2(x + 1)/x + (-1)(x + 1)²/x²
Combining the terms, we have:
dy/dx = [2(x + 1) - (x + 1)²]/x²
Simplifying the expression inside the bracket, we get:
dy/dx = [2x + 2 - x² - 2x - 1]/x²
This simplifies to:
dy/dx = (-x² + 1)/x²
Therefore, the derivative of y = (x + 1)²/x is dy/dx = (-x² + 1)/x².