Find the Derivative - d/dx (x/3.2+3.2/x)(x^2+1)

Problem

d()/d(x)*(x/3.2+3.2/x)*(x2+1)

Solution

1. Expand the expression to simplify the differentiation process.

(x/3.2+3.2/x)*(x2+1)=(x3)/3.2+x/3.2+3.2*x+3.2/x

2. Rewrite the terms using power notation to prepare for the power rule.

1/3.2*x3+1/3.2*x+3.2*x+3.2*x(−1)

3. Combine like terms involving x

1/3.2=0.3125

0.3125*x+3.2*x=3.5125*x

0.3125*x3+3.5125*x+3.2*x(−1)

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

(d(0.3125)*x3)/d(x)=0.9375*x2

(d(3.5125)*x)/d(x)=3.5125

(d(3.2)*x(−1))/d(x)=−3.2*x(−2)

5. Simplify the final expression by converting negative exponents back to fractions.

0.9375*x2+3.5125−3.2/(x2)

Final Answer

d()/d(x)*(x/3.2+3.2/x)*(x2+1)=0.9375*x2+3.5125−3.2/(x2)

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