# Video: Differentiating Exponential Functions Using the Chain Rule

A particle has a position defined by the equations 𝑥 = 𝑡³ − 5𝑡 and 𝑦 = 3 − 2𝑡². Find the velocity vector of the particle at 𝑡 = 2.

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### Video Transcript

A particle has a position defined by the equations 𝑥 equals 𝑡 cubed minus five 𝑡 and 𝑦 equals three minus two 𝑡 squared. Find the velocity vector of the particle at 𝑡 equals two.

We’re told that the position of the particle is defined by a pair of parametric equations. In real terms, this means that, given a value of time 𝑡, we obtain a coordinate pair 𝑥𝑦 for the position of the particle at that time. We could say that, in vector terms, the position of the particle at time 𝑡 is 𝑠 of 𝑡 equals 𝑡 cubed minus five 𝑡 𝑖 plus three minus two 𝑡 squared 𝑗. This isn’t though quite what we’re looking for. We want to find the velocity vector of our particle when 𝑡 is equal to two. So we recall that, given a function for displacement which of course is essentially the difference in the object’s position from one time to another, we find a function for velocity by differentiating with respect to 𝑡.

This means that we can achieve a velocity vector for our particle at time 𝑡 by differentiating each component for the position vector with respect to 𝑡. That’s the derivative of 𝑡 cubed minus five 𝑡. And then we differentiate the vertical component as well, the derivative of three minus two 𝑡 squared. We know that to differentiate a polynomial term, we multiply the entire term by the experiment and then reduce the exposure by one. So 𝑡 cubed differentiates to three 𝑡 squared and negative five 𝑡 differentiates to negative five.

Similarly, the derivative of three minus two 𝑡 squared is negative four 𝑡. Remember, the derivative of any constant is simply zero. Our vector function for velocity is three 𝑡 squared minus five 𝑖 plus negative four 𝑡 𝑗. Remember, we want a velocity vector at 𝑡 is equal to two. So we’re going to substitute 𝑡 equals two into our vector. That’s three times two squared minus five 𝑖 plus negative four times two 𝑗, which simplifies to seven 𝑖 minus eight 𝑗.