What is the derivative of #f(t) = (tlnt, 3t^2+5t ) #?

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Answer 1 · 2017-10-08T11:43:36

# f'(t) = (6t+5)/(1+lnt) #

We have a parametric function of two variables, #x# and #y#, say:

# f(t) = { ( \ x(t),=tlnt), ( \ y(t),=3t^2+5t) :} #

Differentiating wrt #t# we have (via the product rule):

# dx/dt = t(d/dtlnt) + (d/dtt)lnt #
# \ \ \ \ \ = t(1/t) + (1)lnt #
# \ \ \ \ \ = 1 + lnt #

And:

# dy/dt = 6t+5 #

Finally, by the chain rule:

# f'(t) = dy/dx #

# \ \ \ \ \ \ \ \ = (dy//dt) / (dx//dt) #

# \ \ \ \ \ \ \ \ = (6t+5)/(1+lnt) #