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

Problem

d()/d(x)*(x2−4*x−6)*(x3−5*x2−3*x)

Solution

1. Identify the product rule formula for two functions u and v which is d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions u=x2−4*x−6 and v=x3−5*x2−3*x

3. Differentiate u to find d(u)/d(x)=2*x−4

4. Differentiate v to find d(v)/d(x)=3*x2−10*x−3

5. Substitute these components into the product rule formula.

y′=(x2−4*x−6)*(3*x2−10*x−3)+(x3−5*x2−3*x)*(2*x−4)

6. Expand the first part of the expression using distribution.

(x2−4*x−6)*(3*x2−10*x−3)=3*x4−10*x3−3*x2−12*x3+40*x2+12*x−18*x2+60*x+18

7. Simplify the first part by combining like terms.

3*x4−22*x3+19*x2+72*x+18

8. Expand the second part of the expression.

(x3−5*x2−3*x)*(2*x−4)=2*x4−4*x3−10*x3+20*x2−6*x2+12*x

9. Simplify the second part by combining like terms.

2*x4−14*x3+14*x2+12*x

10. Combine both simplified parts to find the final derivative.

y′=(3*x4−22*x3+19*x2+72*x+18)+(2*x4−14*x3+14*x2+12*x)

y′=5*x4−36*x3+33*x2+84*x+18

Final Answer

(d(x2−4*x−6)*(x3−5*x2−3*x))/d(x)=5*x4−36*x3+33*x2+84*x+18

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