Find the Derivative - d/dx (x/3.4+3.4/x)(x^2+1)

Problem

d()/d(x)*(x/3.4+3.4/x)*(x2+1)

Solution

1. Expand the expression by multiplying the two binomials to simplify the differentiation process.

(x/3.4+3.4/x)*(x2+1)=(x3)/3.4+x/3.4+3.4*x+3.4/x

2. Rewrite the terms using power notation to prepare for the power rule, specifically converting the fractions and the term with x in the denominator.

1/3.4*x3+1/3.4*x+3.4*x+3.4*x(−1)

3. Combine the like terms involving x by adding their coefficients.

1/3.4≈0.2941176

1/3.4*x+3.4*x=(1/3.4+3.4)*x

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

d()/d(x)(x3)/3.4=(3*x2)/3.4

d()/d(x)*(1/3.4+3.4)*x=1/3.4+3.4

d()/d(x)*3.4*x(−1)=−3.4*x(−2)

5. Simplify the resulting derivative by combining the terms back into a single expression and converting x(−2) back to a fraction.

(3*x2)/3.4+1/3.4+3.4−3.4/(x2)

Final Answer

d()/d(x)*(x/3.4+3.4/x)*(x2+1)=(3*x2+1)/3.4+3.4−3.4/(x2)

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