Find the Derivative - d/d@VAR f(x)=(4x^4+7x^3-3x^2-4x+5)/(x^3)

Problem

d()/d(x)(4*x^4+7*x^3−3*x^2−4*x+5)/(x^3)

Solution

1. Simplify the expression by dividing each term in the numerator by the denominator x^3

ƒ(x)=(4*x^4)/(x^3)+(7*x^3)/(x^3)−(3*x^2)/(x^3)−(4*x)/(x^3)+5/(x^3)

2. Rewrite the terms using negative exponents to prepare for the power rule.

ƒ(x)=4*x+7−3*x^(−1)−4*x^(−2)+5*x^(−3)

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

d(ƒ)/d(x)=(d(4)*x)/d(x)+d(7)/d(x)−(d(3)*x^(−1))/d(x)−(d(4)*x^(−2))/d(x)+(d(5)*x^(−3))/d(x)

4. Calculate the derivative of each term.

d(ƒ)/d(x)=4+0−3*(−1)*x^(−2)−4*(−2)*x^(−3)+5*(−3)*x^(−4)

5. Simplify the coefficients and signs.

d(ƒ)/d(x)=4+3*x^(−2)+8*x^(−3)−15*x^(−4)

6. Convert the negative exponents back into fraction form.

d(ƒ)/d(x)=4+3/(x^2)+8/(x^3)−15/(x^4)

Final Answer

d()/d(x)(4*x^4+7*x^3−3*x^2−4*x+5)/(x^3)=4+3/(x^2)+8/(x^3)−15/(x^4)

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