Find the Derivative - d/d@VAR F(x) = square root of x-5 cube root of x

Problem

d()/d(x)*(√(,x)−5√(3,x))

Solution

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

√(,x)=x(1/2)

5√(3,x)=5*x(1/3)

2. Apply the power rule d(xn)/d(x)=n*x(n−1) to each term of the expression.

d(x(1/2))/d(x)=1/2*x(−1/2)

(d(5)*x(1/3))/d(x)=5⋅1/3*x(−2/3)

3. Simplify the coefficients and rewrite the terms with positive exponents in the denominator.

1/2*x(−1/2)=1/(2√(,x))

5/3*x(−2/3)=5/(3√(3,x2))

4. Combine the results to find the final derivative of the function.

d()/d(x)*(x(1/2)−5*x(1/3))=1/(2√(,x))−5/(3√(3,x2))

Final Answer

(d(√(,x))−5√(3,x))/d(x)=1/(2√(,x))−5/(3√(3,x2))

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