### Video Transcript

Suppose that π is
differentiable. What is the derivative of π₯ cubed
π of π₯?

So if we take a look at π₯ cubed π
of π₯, what this is is π₯ cubed multiplied by a function. So how are we going to
differentiate this? Well to differentiate this, what
weβre gonna use is something called the product rule. And weβre gonna use that because
the product rule tells us that if we have π¦ is equal to π’π£, so if weβre looking
at two things multiplied together like here we have π₯ cubed and π of π₯, then the
derivative or ππ¦ ππ₯ is gonna be equal to π’ ππ£ ππ₯ plus π£ ππ’ ππ₯. So ππ’ multiplied by the
derivative of π£ plus π£ multiplied by the derivative of π’.

So in our expression, what we have
is π₯ cubed, this is gonna be our π’, and π of π₯, this is gonna be our π£. Therefore, if π’ was equal to π₯
cubed, ππ’ ππ₯ is gonna be equal to three π₯ squared. And just to remind us how we got
that, we got the exponent, which is three, and then we multiply it by the
coefficient, which is one, so three multiplied by one. And then weβve got π₯ to the power
of, and then you subtract one from the exponent. So three minus one gives us two, so
it gives us our three π₯ squared. And then if π£ is our function of
π₯, then ππ£ ππ₯ is gonna be the derivative of this function of π₯, which Iβve
shown using π prime of π₯.

So this is gonna give us the
derivative is equal to π₯ cubed π prime of π₯ or the derivative of π of π₯. And thatβs because that was our π’
multiplied by our ππ£ ππ₯ plus π£ ππ’ ππ₯, so three π₯ squared multiplied by π
of π₯. And thatβs because our π£ was π of
π₯ and our ππ’ ππ₯ was three π₯ squared. And therefore, we can write this
the other way round if we want to have it in increasing powers of π₯. But we can suppose that if π is
differentiable, the derivative of π₯ cubed π of π₯ is going to be three π₯ squared
π of π₯ plus π₯ cubed of the derivative of π of π₯.