How To Differentiate The Product Of Two Functions: A Step-By-Step Guide Using The Product Rule Formula

d/dx[uv]

u’v+uv’

To differentiate the product of two functions u(x) and v(x), we use the product rule, which states:

d/dx [u(x)v(x)] = u(x) d/dx [v(x)] + v(x) d/dx [u(x)]

Using this rule, we can differentiate the product of u and v as follows:

d/dx [uv] = u d/dx [v] + v d/dx [u]

Note that we don’t need to specify whether u or v is the first or second function in this formula since multiplication is commutative.

Now, we need to differentiate u(x) and v(x) separately to solve this derivative. For example, if u(x) = x^2 and v(x) = sin(x), then:

d/dx [u] = d/dx [x^2] = 2x
d/dx [v] = d/dx [sin(x)] = cos(x)

Plugging these derivatives into the product rule formula, we get:

d/dx [uv] = u d/dx [v] + v d/dx [u]
d/dx [uv] = (x^2)(cos(x)) + (sin(x))(2x)

Therefore, the derivative of the product uv with respect to x is (x^2)(cos(x)) + (sin(x))(2x).

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