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

Problem

d()/d(x)(x^3−3*x^2+4)/(x^2)

Solution

1. Simplify the expression by dividing each term in the numerator by the denominator x^2

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

2. Rewrite the terms using exponent rules to make differentiation easier.

ƒ(x)=x−3+4*x^(−2)

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

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 term.

d(ƒ(x))/d(x)=1−0+4*(−2)*x^(−3)

5. Simplify the resulting expression and rewrite with positive exponents.

d(ƒ(x))/d(x)=1−8*x^(−3)

d(ƒ(x))/d(x)=1−8/(x^3)

6. Combine into a single fraction by finding a common denominator.

d(ƒ(x))/d(x)=(x^3−8)/(x^3)

Final Answer

d()/d(x)(x^3−3*x^2+4)/(x^2)=(x^3−8)/(x^3)

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