Product Rule

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Product rule is a formula which is used to find the derivatives of products of two or more functions. If u(x) and v(x) are functions, continuous in an interval [p, q] with derivatives u1(a) and v1(a) respectively at the point x = p, then the derivative of the product y(x) = u(x).v(x) at that point is
y1(p) = u(p) v1(p) + v(p) u1(p)
It is used to find the derivative of two or more functions.
If the equation is y = uvw then
dydx = uw dvdx + vw dudx + uv dwdx
Example 1: Differentiate y= 3x2.e2x
Answer: Here u = 3x2 and v = e2x
dydx = u dvdx + v dudx
dydx = 3x2 . ( d )dx (e2x) + e2x ddx (3x²)
dydx = 3x2.(2e2x) + (e2x) (6x)
dydx = 6x2.e2x + 6x e2x
dydx = 6xe2x (x+1)
Example 2: Differentiate y = (3x3 + 4x2 + 2) (5x3 +9x)
Answer: If y = uv
dydx = u dvdx + v dudx
Here u = 3x3 + 4x2+ 2 and v = 5x3 + 9x
dydx = (3x3 + 4x2 + 2) d/dx (5x3 +9x) + (5x3 +9x). d/dx (3x3 + 4x2 +2)
dydx = (3x3 + 4x2 + 2) (15x2 + 9) + (5x3 +9x). (9x2 + 8x)
dydx=45x^5 + 27x3 + 60x4 +36x2 +30x2 +18 +45x5 + 40x4 +81x3 +72x2
dydx= 90x5 +100x4 +108x3 + 138x2 +18

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