Question Statement
Find the derivative of the function:
y=x3−xx2−10x+2
with respect to x.
Background and Explanation
This problem involves differentiating a rational function, which requires the quotient rule. You should be comfortable with the power rule for polynomials and careful algebraic expansion when simplifying the numerator.
Solution
We are given the rational function:
y=x3−xx2−10x+2
To differentiate using the quotient rule, we identify:
- Numerator: u=x2−10x+2
- Denominator: v=x3−x
First, compute the derivatives of u and v using the power rule:
dxdu=2x−10
dxdv=3x2−1
Applying the quotient rule formula dxdy=v2v⋅dxdu−u⋅dxdv:
dxdy=(x3−x)2(x3−x)(2x−10)−(x2−10x+2)(3x2−1)
Now expand the products in the numerator:
First product: (x3−x)(2x−10)
=x3(2x−10)−x(2x−10)
=2x4−10x3−2x2+10x
Second product: (x2−10x+2)(3x2−1)
=x2(3x2−1)−10x(3x2−1)+2(3x2−1)
=3x4−x2−30x3+10x+6x2−2
=3x4−30x3+5x2+10x−2
Substitute these expansions back into the derivative expression:
dxdy=(x3−x)2(2x4−10x3−2x2+10x)−(3x4−30x3+5x2+10x−2)
Distribute the negative sign through the second set of parentheses:
=(x3−x)22x4−10x3−2x2+10x−3x4+30x3−5x2−10x+2
Combine like terms in the numerator:
- x4 terms: 2−3=−1
- x3 terms: −10+30=+20
- x2 terms: −2−5=−7
- x terms: +10−10=0
- Constants: +2
Therefore, the final derivative is:
dxdy=(x3−x)2−x4+20x3−7x2+2
- Quotient Rule: dxd(vu)=v2v⋅dxdu−u⋅dxdv
- Power Rule: dxd(xn)=nxn−1
- Polynomial expansion and combining like terms for algebraic simplification
Summary of Steps
- Identify the numerator u=x2−10x+2 and denominator v=x3−x
- Differentiate both using the power rule: u′=2x−10 and v′=3x2−1
- Apply the quotient rule formula: v2vu′−uv′
- Expand both products in the numerator carefully
- Distribute the negative sign and combine like terms
- Simplify to obtain the final result: (x3−x)2−x4+20x3−7x2+2