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

Problem

d()/d(x)(4*x4+7*x3−3*x2−4*x+5)/(x3)

Solution

1. Simplify the expression by dividing each term in the numerator by the denominator x3

ƒ(x)=(4*x4)/(x3)+(7*x3)/(x3)−(3*x2)/(x3)−(4*x)/(x3)+5/(x3)

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(xn)/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/(x2)+8/(x3)−15/(x4)

Final Answer

d()/d(x)(4*x4+7*x3−3*x2−4*x+5)/(x3)=4+3/(x2)+8/(x3)−15/(x4)

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