Find the Derivative - d/dx y = cube root of 1+5x

Problem

d()/d(x)√(3,1+5*x)

Solution

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

√(3,1+5*x)=(1+5*x)(1/3)

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

d(1+5*x)/d(x)=1/3*(1+5*x)(1/3−1)⋅d(1+5*x)/d(x)

3. Differentiate the inner function 1+5*x with respect to x

d(1+5*x)/d(x)=5

4. Substitute the inner derivative back into the expression and simplify the exponent.

1/3*(1+5*x)(−2/3)⋅5

5. Simplify the final expression by multiplying the constants and moving the negative exponent to the denominator.

5/(3*(1+5*x)(2/3))

Final Answer

d(√(3,1+5*x))/d(x)=5/(3√(3,(1+5*x)2))

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