Home > Calculus > Derivative of Parametric Functions

What is the derivative of #f(t) = (t^3-e^(3t-1) , -t^2-e^t ) #?

Answer 1

Samantha Adkison

Derivative is #-(2t+e^t)/(3t^2-3e^(3t-1))#

The derivative of a function #f(t)=(x(t),y(t))# is given by #((dy)/(dt))/((dx)/(dt))#

Here #x(t)=t^3-e^(3t-1)# and hence

#(dx)/(dt)=3t^2-e^(3t-1)*3=3t^2-3e^(3t-1)#

and as #y(t)=-t^2-e^t#, #(dy)/(dt)=-2t-e^t#

Hence derivative is #(-2t-e^t)/(3t^2-3e^(3t-1))=-(2t+e^t)/(3t^2-3e^(3t-1))#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Samantha Adkison

The derivative of (f(t) = (t^3 - e^{3t - 1}, -t^2 - e^t)) with respect to (t) is given by:

[\frac{d}{dt}f(t) = \left(\frac{d}{dt}(t^3 - e^{3t - 1}), \frac{d}{dt}(-t^2 - e^t)\right)]

Solving each component separately:

[\frac{d}{dt}(t^3 - e^{3t - 1}) = 3t^2 - 3e^{3t - 1}]

[\frac{d}{dt}(-t^2 - e^t) = -2t - e^t]

Therefore, the derivative of (f(t)) with respect to (t) is:

[\frac{d}{dt}f(t) = (3t^2 - 3e^{3t - 1}, -2t - e^t)]

By signing up, you agree to our Terms of Service and Privacy Policy

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.