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

Problem

d()/d(x)*x√(3,x+1)

Solution

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

y=x*(x+1)(1/3)

2. Apply the product rule, which states that d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=x and v=(x+1)(1/3)

d(y)/d(x)=xd(x+1)/d(x)+(x+1)(1/3)d(x)/d(x)

3. Apply the chain rule to differentiate (x+1)(1/3) and the power rule to differentiate x

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

4. Simplify the expression by writing the terms with a common denominator.

d(y)/d(x)=x/(3*(x+1)(2/3))+(x+1)(1/3)

5. Find a common denominator of 3*(x+1)(2/3) to combine the terms.

d(y)/d(x)=(x+3*(x+1)(2/3)*(x+1)(1/3))/(3*(x+1)(2/3))

6. Simplify the numerator by adding the exponents of the (x+1) terms.

d(y)/d(x)=(x+3*(x+1))/(3*(x+1)(2/3))

7. Distribute and combine like terms in the numerator.

d(y)/d(x)=(x+3*x+3)/(3*(x+1)(2/3))

8. Finalize the expression.

d(y)/d(x)=(4*x+3)/(3√(3,(x+1)2))

Final Answer

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

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