Differentiate the function f(x) = sin(x)/(x^2 +1) , giving your answer in the form of a single fraction. Is x=0 a stationary point of this curve?

The key concepts to apply in this question will be the product and chain rules, namely: if  f(x)=g(x)h(h), then f'(x)=g(x)h'(x) + g'(x)h(x), and if h(x)=u(v(x)), then h'(x)=u'(v(x))v'(x).

Equivalently, you may prefer to apply the quotient rule and the chain rule.

To answer this question, you also need to know that x is a stationary point if f'(x)=0.

Worked solution:

Here we have g(x)=sin(x) and h(x)=(x²+1)^-1. We differentiate these to get g'(x)=cos(x) and h'(x)=(-1)(2x)(x²+1)^-2, using the chain rule to differentiate h(x).

Now we put these together (using the product rule) to get f'(x)=sin(x)(-1)(2x)(x²+1)^-2+cos(x)(x²+1)^-1.

Finally, the question asks for the final answer in the form of a single fraction, so we rearrange to get: f'(x)=(cos(x)*(x²+1) - (2x)*sin(x))/(x²+1)².

To finish off we need to check the value of f'(x) at =0: f'(0)=1/1²=1. This is not 0, so x=0 is not a stationary point.