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

Problem

d()/d(x)(x3−3*x2+4)/(x2)

Solution

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

ƒ(x)=(x3)/(x2)−(3*x2)/(x2)+4/(x2)

2. Rewrite the terms using exponent rules to prepare for differentiation.

ƒ(x)=x−3+4*x(−2)

3. Apply the power rule, which states d(xn)/d(x)=n*x(n−1) to each individual term.

d(ƒ(x))/d(x)=d(x)/d(x)−d(3)/d(x)+(d(4)*x(−2))/d(x)

4. Calculate the derivative of each part: the derivative of x is 1 the derivative of a constant is 0 and the derivative of 4*x(−2) is −8*x(−3)

d(ƒ(x))/d(x)=1−0−8*x(−3)

5. Rewrite the result using positive exponents for the final form.

d(ƒ(x))/d(x)=1−8/(x3)

Final Answer

d()/d(x)(x3−3*x2+4)/(x2)=1−8/(x3)

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