Find the Derivative - d/dx y=(3x^2-4x+1)(5-6x^2)

Problem

d()/d(x)*(3*x2−4*x+1)*(5−6*x2)

Solution

1. Identify the product rule formula, which states that for a function y=u⋅v the derivative is d(y)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the expression to u and v

u=3*x2−4*x+1

v=5−6*x2

3. Differentiate each part individually:

d(u)/d(x)=6*x−4

d(v)/d(x)=−12*x

4. Substitute these into the product rule formula:

d(y)/d(x)=(3*x2−4*x+1)*(−12*x)+(5−6*x2)*(6*x−4)

5. Expand the first term by distributing −12*x

(3*x2−4*x+1)*(−12*x)=−36*x3+48*x2−12*x

6. Expand the second term using the FOIL method:

(5−6*x2)*(6*x−4)=30*x−20−36*x3+24*x2

7. Combine the expanded terms and group like powers of x

d(y)/d(x)=−36*x3+48*x2−12*x−36*x3+24*x2+30*x−20

8. Simplify by adding the coefficients of the like terms:

d(y)/d(x)=−72*x3+72*x2+18*x−20

Final Answer

d()/d(x)*(3*x2−4*x+1)*(5−6*x2)=−72*x3+72*x2+18*x−20

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