Find the Derivative - d/dx

Problem

d()/d(x)1/(4√(,x)+x√(,x))

Solution

1. Rewrite the expression using exponent notation to make differentiation easier.

x√(,x)=x1⋅x(1/2)=x(3/2)

1/(4*x(1/2)+x(3/2))=(4*x(1/2)+x(3/2))(−1)

2. Apply the chain rule by letting the inner function be u=4*x(1/2)+x(3/2)

d(u(−1))/d(x)=−u(−2)⋅d(u)/d(x)

3. Differentiate the inner function using the power rule.

d(4*x(1/2)+x(3/2))/d(x)=4⋅1/2*x(−1/2)+3/2*x(1/2)

d(u)/d(x)=2*x(−1/2)+3/2*x(1/2)

4. Combine the results and substitute u back into the expression.

−(4*x(1/2)+x(3/2))(−2)⋅(2*x(−1/2)+3/2*x(1/2))

5. Simplify the expression by moving the negative exponent to the denominator and factoring.

−(2*x(−1/2)+3/2*x(1/2))/((4*x(1/2)+x(3/2))2)

−(4+3*x)/(2√(,x))/((√(,x)*(4+x))2)

−(4+3*x)/(2√(,x)⋅x*(4+x)2)

−(4+3*x)/(2*x(3/2)*(4+x)2)

Final Answer

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

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