Find the Derivative - d/dx (3x+2 square root of x+32/(x^2))

Problem

d()/d(x)*(3*x+2√(,x)+32/(x2))

Solution

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

3*x+2*x(1/2)+32*x(−2)

2. Apply the sum rule for derivatives, which allows for differentiating each term individually.

(d(3)*x)/d(x)+(d(2)*x(1/2))/d(x)+(d(32)*x(−2))/d(x)

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

3*(1)*x0+2*(1/2)*x(−1/2)+32*(−2)*x(−3)

4. Simplify the coefficients and exponents.

3+x(−1/2)−64*x(−3)

5. Convert the negative and fractional exponents back into radical and fraction form.

3+1/√(,x)−64/(x3)

Final Answer

d()/d(x)*(3*x+2√(,x)+32/(x2))=3+1/√(,x)−64/(x3)

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