Differentiate each of the following functions by the product by the product rule and the other method and verify that answer from both the methods is the same.
(x + 2)(x + 3)
Let, y = (x + 2)(x + 3)
⇒ y = x2 + 5x + 6
Differentiating y w.r.t x –
Using algebra of derivatives, we have –
Use formula of derivative of above function to get the result.
⇒ 
{∵ 
∴ 
…equation 1
Derivative using product rule –
We have to find dy/dx
As we can observe that y is a product of two functions say u and v where,
u = (x + 2) and v = (x + 3)
∴ y = uv
As we know that to find the derivative of product of two function we apply product rule of differentiation.
By product rule, we have –
…equation 2
As, u = (x + 2)
∴ 
⇒ 
⇒ 
…..equation 3 {∵ 
}
As, v = (x + 3)
⇒ 
⇒ 
…equation 4 {∵ 
}
∴ from equation 2, we can find dy/dx
∴ 
using equation 3 & 4, we get –
⇒ 
⇒ 
Hence,
….equation 5
Clearly from equation 1 and 5 we observed that both equations gave identical results.
Hence, Results are verified