Find the Derivative - d/dx (x^3+x+4)/(2x)

Problem

d over d x the fraction with numerator x cubed plus x plus 4 and denominator 2 x

Solution

1. Simplify the expression by dividing each term in the numerator by the denominator to avoid the quotient rule.

the fraction with numerator x cubed plus x plus 4 and denominator 2 x equals the fraction with numerator x cubed and denominator 2 x plus x over 2 x plus 4 over 2 x

2. Reduce the powers of x and rewrite the terms using negative exponents where necessary.

one half x squared plus one half plus 2 x to the negative 1 power

3. Apply the power rule, which states d x to the n-th power over d x equals n x raised to the n minus 1 power to each individual term.

d over d x one half x squared equals x

d over d x one half equals 0

d over d x 2 x to the negative 1 power equals negative 2 x to the negative 2 power

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

x minus the fraction with numerator 2 and denominator x squared

5. Find a common denominator to express the final result as a single fraction.

the fraction with numerator x cubed minus 2 and denominator x squared

Final Answer

d over d x the fraction with numerator x cubed plus x plus 4 and denominator 2 x equals the fraction with numerator x cubed minus 2 and denominator x squared

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