Product Rule

A curve is defined by the equation $y=(x^2+3x+2)\sin x$.

Find the gradient of the tangent to this curve when $x=0$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Find the $x$-value(s) of any points of inflection on the graph of the function $f(x)=3x^2\ln (x)$.

You can assume that your point(s) found are actually point(s) of inflection.

You must use calculus and show any derivatives that you need to find when solving this problem.

A curve has the equation $y=x^2\cos x$.

Show that the equation of the tangent to the curve at the point $(\pi ,-{\pi }^2)$ is

$y+2\pi x={\pi }^2$

*You must use calculus and show any derivatives that you need to find when solving this problem.*

For what value(s) of $x$ is the function $y=x^3{e}^x$ decreasing?

You must use calculus and show any derivatives that you need to find when solving this
problem.

Find the equation of the tangent to the curve $y=x^2\ln x$ at the point where $x=e$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

If $y=e^x(2x^2-x-1)$, find the value(s) of $x$ for which $\frac{dy}{dx}=0$.

You must use calculus and show any derivatives that you need to find when solving this
problem.

For what values of $x$ is the function $f(x)=5x-x\ln x$ increasing?

Show any derivatives that you need to find when solving this problem.