9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

Guida verificata passo dopo passo

First, identify the function y given in the problem. Here, y = 2x² cos⁻¹(x) + sin⁻¹(x).

To find y', the derivative of y with respect to x, apply the derivative rules to each term separately. Start with the term 2x² cos⁻¹(x).

For the term 2x² cos⁻¹(x), use the product rule: if u = 2x² and v = cos⁻¹(x), then y' = u'v + uv'. Calculate u' and v' separately.

The derivative of u = 2x² is u' = 4x. The derivative of v = cos⁻¹(x) is v' = -1/√(1-x²). Substitute these into the product rule formula.

Next, find the derivative of the second term sin⁻¹(x). The derivative of sin⁻¹(x) is 1/√(1-x²). Combine the derivatives of both terms to get y'.

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