Find the Derivative - d/dx x(x^2+5)^(1/3)

Problem

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

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=(x^2+5)^(1/3)

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

3. Differentiate the first part u=x to get d(u)/d(x)=1

4. Differentiate the second part v=(x^2+5)^(1/3) using the chain rule.

5. Calculate the derivative of the second part: d(x^2+5)/d(x)=1/3*(x^2+5)^(−2/3)⋅2*x

6. Substitute these components back into the product rule formula.

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

7. Simplify the expression by finding a common denominator, which is 3*(x^2+5)^(2/3)

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

8. Combine the numerators to reach the final simplified form.

(2*x^2+3*x^2+15)/(3*(x^2+5)^(2/3))

(5*x^2+15)/(3*(x^2+5)^(2/3))

Final Answer

(d(x)*(x^2+5)^(1/3))/d(x)=(5*x^2+15)/(3*(x^2+5)^(2/3))

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