Find the Derivative - d/dx y=(3-x^2)(x^3-x+1)

Problem

d()/d(x)*(3−x2)*(x3−x+1)

Solution

1. Identify the function as a product of two terms, u=3−x2 and v=x3−x+1 which requires the product rule.

2. Apply the product rule formula, which states d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual components: d(3−x2)/d(x)=−2*x and d(x3−x+1)/d(x)=3*x2−1

4. Substitute these derivatives back into the product rule formula:

(3−x2)*(3*x2−1)+(x3−x+1)*(−2*x)

5. Expand the first part of the expression using the FOIL method:

9*x2−3−3*x4+x2

6. Expand the second part of the expression by distributing −2*x

−2*x4+2*x2−2*x

7. Combine all terms and group like powers of x

−3*x4−2*x4+9*x2+x2+2*x2−2*x−3

8. Simplify the expression to find the final derivative:

−5*x4+12*x2−2*x−3

Final Answer

d()/d(x)*(3−x2)*(x3−x+1)=−5*x4+12*x2−2*x−3

Want more problems? Check here!