Find the Derivative - d/dx (x/3.5+3.5/x)(x^2+1)

Problem

d()/d(x)*(x/3.5+3.5/x)*(x2+1)

Solution

1. Rewrite the expression by converting the decimal 3.5 to the fraction 7/2 to make calculations easier.

((2*x)/7+7/(2*x))*(x2+1)

2. Expand the product using the FOIL method before differentiating.

(2*x)/7*(x2)+(2*x)/7*(1)+7/(2*x)*(x2)+7/(2*x)*(1)

3. Simplify each term in the expanded expression.

(2*x3)/7+(2*x)/7+(7*x)/2+7/(2*x)

4. Combine the middle terms with a common denominator of 14

(2*x3)/7+(4*x+49*x)/14+7/2*x(−1)

(2*x3)/7+(53*x)/14+7/2*x(−1)

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

d()/d(x)(2*x3)/7=(6*x2)/7

d()/d(x)(53*x)/14=53/14

d()/d(x)7/2*x(−1)=−7/2*x(−2)

6. Combine the results and rewrite the negative exponent as a fraction.

(6*x2)/7+53/14−7/(2*x2)

Final Answer

d()/d(x)*(x/3.5+3.5/x)*(x2+1)=(6*x2)/7+53/14−7/(2*x2)

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