Find the Derivative - d/dx square root of x-2 cube root of x

Problem

d()/d(x)*(√(,x)−2√(3,x))

Solution

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

√(,x)=x(1/2)

2√(3,x)=2*x(1/3)

2. Apply the sum/difference rule for derivatives, which allows for differentiating each term independently.

d()/d(x)*(x(1/2)−2*x(1/3))=d(x(1/2))/d(x)−(d(2)*x(1/3))/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)

(d(2)*x(1/3))/d(x)=2⋅1/3*x(−2/3)

4. Simplify the coefficients and rewrite the negative exponents as denominators.

1/2*x(−1/2)=1/(2√(,x))

2/3*x(−2/3)=2/(3√(3,x2))

Final Answer

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

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