Question

calculate the derivative of x^3cosx

Answer 1

Since g(x) is the product of two terms, we can use the Product Rule to find the derivative.

We essentially have

$g(x)=f(x)\cdot h(x)$

where,

$\begin{gathered} f(x)=x^3\text{ and} \\ h(x)=cosx \end{gathered}$

Thus, the Product Rule states that the derivative is equal to:

$f^{\prime}(x)h(x)+f(x)h^{\prime}(x)$

To differentiate f(x), we can use the Power Rule, where the exponent becomes the coefficient, and we decrement the power.

Therefore,

$f^{\prime}(x)=3x^2$

And from our knowledge of derivatives of trig functions

$h^{\prime}(x)=-\sin x$

We can now plug these values into the product rule expression to get

$3x^2(\cos x)+x^3(-\sin x)$

We can rewrite this as

$3x^2(\cos x)-x^3(\sin x)$

Hence,

$3x^2\cos x-x^3\sin x$

Therefore, the derivative is

$3x^2\cos x-x^3\sin x$