Question

Which of the following expressions can be differentiated using the product rule?

A) x² ⋅sinx
B) ln(x)⋅ex

C) ex⋅cosx
D) sin(x)⋅cos(x)

Answer 1

All expressions A) x² ⋅sinx, B) ln(x)⋅eˣ, C) eˣ⋅cosx, D) sin(x)⋅cos(x) can be differentiated using the product rule, which requires differentiating each function separately and then combining the results as specified by the rule.

Step-by-step explanation:

The question asks which of the given expressions can be differentiated using the product rule. The product rule is a derivative rule used for functions that are the products of two other functions. Each of the expressions given (A) x² ⋅sinx, (B) ln(x)⋅eˣ, (C) eˣ⋅cosx, and (D) sin(x)⋅cos(x), are products of two functions. Therefore, all of them can be differentiated using the product rule.

The product rule states that for two differentiable functions, u(x) and v(x), the derivative of their product u(x)v(x) is given by:

u'(x)v(x) + u(x)v'(x)

When applying the product rule:

• First, take the derivative of the first function (u'(x)) and multiply by the second function (v(x)).
• Then, take the derivative of the second function (v'(x)) and multiply by the first function (u(x)).
• Finally, add the two results together.

Examples when applying the product rule:

1. For x² ⋅sinx, differentiate x² to get 2x and differentiate sinx to get cosx, then apply the product rule.
2. For ln(x)⋅eˣ, differentiate ln(x) to get 1/x and differentiate eˣ to get eˣ, then apply the product rule.
3. For eˣ⋅cosx, differentiate eˣ to get eˣ and differentiate cosx to get -sinx, then apply the product rule.
4. For sin(x)⋅cos(x), differentiate sin(x) to get cos(x) and differentiate cos(x) to get -sin(x), then apply the product rule.