Find the Derivative - d/dx y = cube root of 6x^2+1

Problem

d()/d(x)√(3,6*x2+1)

Solution

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

√(3,6*x2+1)=(6*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(6*x2+1)/d(x)=1/3*(6*x2+1)(−2/3)⋅d(6*x2+1)/d(x)

3. Differentiate the inner function 6*x2+1 using the power rule.

d(6*x2+1)/d(x)=12*x

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

1/3*(6*x2+1)(−2/3)⋅12*x

5. Simplify the expression by multiplying the constants and rewriting the negative exponent as a denominator.

(12*x)/(3*(6*x2+1)(2/3))=(4*x)/((6*x2+1)(2/3))

Final Answer

d(√(3,6*x2+1))/d(x)=(4*x)/√(3,(6*x2+1)2)

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