How do you find the expression dydx when given the equation yt = sin(t) + t^2 + 3t?

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To find the derivative dy/dx for the given equation, we first need to rewrite the expression in terms of known variables. The function provided is y(t) = sin(t) + t2 + 3t. This means we need to differentiate y with respect to t.

The process of differentiation involves applying the rules of calculus:

  • Derivative of sin(t): The derivative of sin(t) is cos(t).
  • Derivative of t2: The derivative of t squared is 2t.
  • Derivative of 3t: The derivative of 3t is simply 3.

So, applying the differentiation step by step, we have:

y'(t) = d/dt[sin(t) + t2 + 3t] 
          = cos(t) + 2t + 3

Now we have the first derivative y'(t) = cos(t) + 2t + 3.

If we want to convert this into dx/dt (which we can denote as v), the relationship known as the chain rule will help:

  • If x is a function of t as well, we can relate dy/dx using the chain rule:
  • dy/dx = (dy/dt) / (dx/dt)

Thus, if we also find dx/dt, we can find the dy/dx more comprehensively:

dy/dx = (cos(t) + 2t + 3) / (dx/dt)

In summary, to find dydx from the given expression y = sin(t) + t2 + 3t, we differentiated y with respect to t, obtaining the derivative:

dy/dt = cos(t) + 2t + 3

Now, utilizing the chain rule, we can relate it to dx/dt. The final expression of dy/dx requires you to know the relationship between x and t to complete the differentiation process.

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