Differentiate t²sin(2t)

Step-by-step explanation:

We are given a function:

To differentiate a function, we are going to use the product rules since there are two functions which are t^2 and sin(2t) multiplying together.

From the product rules,

Therefore,

Before we start differentiating, I’d like you to look at [sin(2t)]’. A function like this, you cannot just directly derive and answer. You need to use chain rules.

We know that, but what if the “x” is another function? Like sin(3x), sin(x^2) as examples. The insides are another function, can be expressed as fog(x) or gof(x) is composite function.

Basically, chain rule is a rule or formula for composite function and it’s the most common and useful as well as being always used in differentiation.

Chain Rules

where u is another function in a bracket. From the formula above, it can be also written as:

To simply say, you differentiate the whole function first then multiply with the chain or inner derived function.

So from the function, we obtain:

The factored form would be:

From above, for polynomial function, to differentiate, you can do by using the power rules.

Power Rules