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We have a "fraction" which we wish to differentiate, so we use the quotient rule with u=sin(x) and v=cos(x).

This means that d/dx of u/v = (vdu/dx - udv/dx)/(v^2).

y = cos(x)/sec²(x) = cos³(x)

y = cos(x)(1-sin²(x)) = cos(x) - cos(x) sin²(x)

-> sin(x) - sin³(x)/3 + c

First let a = b = x such that:          

          cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

becomes:

          cos(x + x) = cos(x)cos(x) - sin(x)sin(x)

Leading to:

     ...

Note that this property is the definition of an odd function, or draw a sketch of what this looks like in general about the horizontal axis. E.g. f(x)=sin(x) which has the expansion f(x)=x-((x^3)/3!)+((x^...

Find the derivative of f(x)=x³ sin(x).

To do this calculation we need to use the product rule of differentiation: if f(x)=u(x)v(x), then the derivative is f'(x)=u'(x)v(x)+u(x)v'(x). In o...