Find the Derivative Using Chain Rule - d/dx y = cube root of 1+4x

Problem

d()/d(x)√(3,1+4*x)

Solution

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

√(3,1+4*x)=(1+4*x)(1/3)

2. Identify the outer function and the inner function for the Chain Rule.

ƒ(u)=u(1/3)

u=1+4*x

3. Apply the Power Rule to the outer function while keeping the inner function the same.

d()/d(u)*u(1/3)=1/3*u(−2/3)

4. Multiply by the derivative of the inner function 1+4*x with respect to x

d()/d(x)*(1+4*x)=4

5. Combine the results using the Chain Rule formula d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

d()/d(x)*(1+4*x)(1/3)=1/3*(1+4*x)(−2/3)⋅4

6. Simplify the expression by multiplying the constants and rewriting the negative exponent.

4/(3*(1+4*x)(2/3))

Final Answer

d(√(3,1+4*x))/d(x)=4/(3√(3,(1+4*x)2))

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