Find the Derivative - d/dx cube root of x^2-1

Problem

d(√(3,x2−1))/d(x)

Solution

1. Rewrite the expression using a rational exponent to make it easier to differentiate.

√(3,x2−1)=(x2−1)(1/3)

2. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

d(x2−1)/d(x)=1/3*(x2−1)(1/3−1)⋅d(x2−1)/d(x)

3. Differentiate the inner function x2−1 with respect to x

d(x2−1)/d(x)=2*x

4. Substitute the derivative of the inner function back into the expression.

1/3*(x2−1)(−2/3)⋅2*x

5. Simplify the expression by combining terms and moving the negative exponent to the denominator.

(2*x)/(3*(x2−1)(2/3))

Final Answer

d(√(3,x2−1))/d(x)=(2*x)/(3√(3,(x2−1)2))

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