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

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

3. Differentiate the first term, 3*x

(d(3)*x)/d(x)=3

4. Differentiate the second term, 2*x(1/2)

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

5. Differentiate the third term, 32*x(−2)

(d(32)*x(−2))/d(x)=32⋅(−2)*x(−3)=−64*x(−3)

6. Simplify the resulting expression by converting negative 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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