Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (x²+1)^x

Guida verificata passo dopo passo

Step 1: For part (i), rewrite the function y = (x²+1)^x using the property b^x = e^(x ln b). This gives y = e^(x ln(x²+1)).

Step 2: Differentiate y = e^(x ln(x²+1)) with respect to x using the chain rule. The derivative of e^u with respect to x is e^u * du/dx, where u = x ln(x²+1).

Step 3: Find du/dx where u = x ln(x²+1). Use the product rule for differentiation: if u = v * w, then du/dx = v' * w + v * w'. Here, v = x and w = ln(x²+1).

Step 4: For part (ii), use logarithmic differentiation. Take the natural logarithm of both sides: ln(y) = x ln(x²+1). Differentiate both sides with respect to x. The left side becomes (1/y) * dy/dx, and the right side is differentiated using the product rule.

Step 5: Solve for dy/dx in both methods and verify that the expressions obtained are equivalent, confirming that both methods yield the same derivative.

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