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First, we need to find the derivative f'(x) for all x, and then we will substitute x=2π into the function we find. To do this, we need to set up our limit: f'(x) = lim [sin(3(x+h))-sin(3x)]/h as h-> 0 We ...

Base Case n=1sum_1(r 3^(r-1))=1 3^0=1=1/4+(3^1/4) (2-1)=1/4+3/4=1 Assume true for n=ki.e. sum_k(r 3^(r-1))=(3^k/4) (2k-1) then for n=k+1sum_(k+1)(r 3^(r-1))= sum_k(r 3^(r-1))+(k+1) 3^k =(3^k/4) (2k-1)+(k+1) ...

This question challenges the student's ability to use the chain rule and to understand what a stationary point is. Having recognised that the chain rule is required, the first step is to use the substitution...

Calculating the integral of ln(x) is harder than it might look and must be done using integration by parts, where the two parts are 1 and ln(x). The integration by parts formula is as follows integral{udv) =...

We’re being asked to differentiate the function with respect to x. To do so, we use the general formula for differentiation, which is: ‘If y = kxn, dy/dx = nkx(n-1) We must also note that, which differentiat...