Differentiate x^3(sinx) with respect to x

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Differentiate x^3(sinx) with respect to x

As we are differentiating a product (two things times together) we can use the product rule which is if:

y = u(x)v(x)

then

dy/dx = u(dv/dx) + v(du/dx).

So firstly looking at our equation we need to identify u(x) and v(x). In our case

u(x) = x³ and v(x) = sinx

Now we need to differentiate both of them seperatly so (remember when we differentiate we times by the old power and then subtract a power)

du/dx = 3x²and dv/dx = cosx

Now putting all this into the formula we have

dy/dx = u(dv/dx) + v(du/dx)

= x³cosx + sinx(3x²)

Then rearranging this we get

dy/dx = x³cosx + 3x²sinx

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