Find the Derivative - d/dx f(x) = square root of x+ fifth root of x

Problem

d()/d(x)*(√(,x)+√(5,x))

Solution

1. Rewrite the radical expressions as powers using the rule √(n,xm)=x(m/n)

√(,x)=x(1/2)

√(5,x)=x(1/5)

2. Apply the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives.

d()/d(x)*(x(1/2)+x(1/5))=d(x(1/2))/d(x)+d(x(1/5))/d(x)

3. Use the power rule d(xn)/d(x)=n*x(n−1) for each term.

d(x(1/2))/d(x)=1/2*x(1/2−1)=1/2*x(−1/2)

d(x(1/5))/d(x)=1/5*x(1/5−1)=1/5*x(−4/5)

4. Simplify the expression by rewriting the negative exponents as denominators.

1/2*x(−1/2)=1/(2√(,x))

1/5*x(−4/5)=1/(5√(5,x4))

Final Answer

d()/d(x)*(√(,x)+√(5,x))=1/(2√(,x))+1/(5√(5,x4))

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