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What is the derivative of #f(t) = (tcos^2t , t^2-t sect ) #?

Answer 1

#(dy)/(dx)=(2t-sect-tsect tant)/(cos^2t-tsin2t)#

When a functiion is given in parametric form such as #f(t)=(x(t),y(t))#, its dervative is given by #(dy)/(dx)=((dy)/(dt))/((dx)/(dt))#

Here we have #y(t)=t^2-tsect# hence #(dy)/(dt)=2t-sect-tsect tant#

and #x(t)=tcos^2t# hence #(dx)/(dt)=cos^2t+txx2costxx(-sint)#

= #cos^2t-tsin2t#

Hence #(dy)/(dx)=(2t-sect-tsect tant)/(cos^2t-tsin2t)#

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Answer 2

Emily Ammerman

To find the derivative of ( f(t) = (t\cos^2(t), t^2 - t\sec(t)) ):

Differentiate each component of ( f(t) ) with respect to ( t ) separately.
For the first component ( t\cos^2(t) ), use the product rule.
For the second component ( t^2 - t\sec(t) ), differentiate each term separately.
Combine the derivatives to form the derivative of the function ( f(t) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.