Use the Product Rule to compute the derivative

Question

(d/dx)((X^2+1)(X+9)) | X=13?

Answer 1

Let y = (x² + 1)(x + 9)

y' = (d/dx)(x² + 1)(x + 9)

Recall the product rule: (d/dx)(UV) = U' V + U V'

y' = (x² + 1)' (x + 9) + (x² + 1)(x + 9)'

General derivative formula: (d/dx)(xⁿ) = nx^n-1 and The Derivative of a Constant is 0

y' = (2x + 0)(x + 9) + (x² + 1)(1+0)

y' = (2x)(x + 9) + (x² + 1)

y' = 2x² + 18x + x² + 1

y' = 3x² + 18x + 1

 

If x = 13 ?

y' = 3(13)² + 18(13) + 1

y' = 3(169) + 234 + 1

y' = 507 + 235

y' = 742

Answer 2

Let y=(x^2+1)(x+9) y = (x^2)(x) +(x^2)(9) +1(x) + 1(9) y = x^3 + 9x^2 + x + 9 Differentiate with respect to x each side dy/dx = d/dx(x^3) + d/dx(9x^2) + d/dx(x) + d/dx(9) Recall : d/dx(x^n) = n*x^n-1 dy/dx = 3x^3-1 + 9*2x^2-1 + 1 +0 dy/dx = 3x^2 + 18x + 1 If x = 13 dy/dx = 3(13)^2 + 18(13) +1 dy/dx = 3(169) + 234 +1 dy/dx = 507 + 235 dy/dx = 742