Find the Derivative - d/dx (x^2-5x+2)(5x^3-x^2+5)

Problem

d()/d(x)*(x2−5*x+2)*(5*x3−x2+5)

Solution

1. Identify the product rule for differentiation, which states that for two functions u and v d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions u=x2−5*x+2 and v=5*x3−x2+5

3. Differentiate u to find d(u)/d(x)=2*x−5

4. Differentiate v to find d(v)/d(x)=15*x2−2*x

5. Substitute these components into the product rule formula.

d()/d(x)*(x2−5*x+2)*(5*x3−x2+5)=(x2−5*x+2)*(15*x2−2*x)+(5*x3−x2+5)*(2*x−5)

6. Expand the first term (x2−5*x+2)*(15*x2−2*x)

(x2−5*x+2)*(15*x2−2*x)=15*x4−2*x3−75*x3+10*x2+30*x2−4*x

(x2−5*x+2)*(15*x2−2*x)=15*x4−77*x3+40*x2−4*x

7. Expand the second term (5*x3−x2+5)*(2*x−5)

(5*x3−x2+5)*(2*x−5)=10*x4−25*x3−2*x3+5*x2+10*x−25

(5*x3−x2+5)*(2*x−5)=10*x4−27*x3+5*x2+10*x−25

8. Combine the expanded terms and simplify by adding like terms.

15*x4−77*x3+40*x2−4*x+10*x4−27*x3+5*x2+10*x−25

25*x4−104*x3+45*x2+6*x−25

Final Answer

d()/d(x)*(x2−5*x+2)*(5*x3−x2+5)=25*x4−104*x3+45*x2+6*x−25

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