Find dy/dx y=(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 by multiplying the two binomials.

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

2. Simplify the terms by reducing the fractions and rewriting the reciprocal term using a negative exponent.

y=1/3.2*x3+1/3.2*x+3.2*x+3.2*x(−1)

3. Combine the like terms involving x to further simplify the expression.

y=1/3.2*x3+(1/3.2+3.2)*x+3.2*x(−1)

4. Calculate the decimal coefficients to make the power rule easier to apply.

1/3.2=0.3125

y=0.3125*x3+3.5125*x+3.2*x(−1)

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

d(y)/d(x)=3*(0.3125)*x2+3.5125−3.2*x(−2)

6. Simplify the final coefficients and rewrite the negative exponent as a fraction.

d(y)/d(x)=0.9375*x2+3.5125−3.2/(x2)

Final Answer

d(y)/d(x)=0.9375*x2+3.5125−3.2/(x2)

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