If 𝑥 equals 𝑡 squared plus four 𝑡 minus two and 𝑦 equals two 𝑡 squared minus three, find d𝑦 d𝑥 at 𝑡 equals one.
Here we have two equations in terms of 𝑡. And we are asked to find d𝑦 d𝑥 at 𝑡 equals one. So once we find d𝑦 d𝑥, we will plug in one. So if this is our equation for 𝑥, we can take it and find the derivative of 𝑥 with respect to 𝑡, d𝑥 d𝑡. And if this is our equation for 𝑦, we can find d𝑦 d𝑡, derivative of 𝑦 with respect to 𝑡.
Now both of these are helpful because if we would take d𝑦 d𝑡 and divide it by d𝑥 d𝑡, this is the same as taking d𝑦 d𝑡 and multiplying by the reciprocal for denominator, d𝑡 d𝑥. The d𝑡s cancel, and we’re left with d𝑦 d𝑥, which is exactly what we’re supposed to find, and then plug in one.
So let’s begin by finding d𝑥 d𝑡, the derivative of 𝑥 with respect to 𝑡. So we need to use the power rule. 𝑡 squared will turn into two 𝑡 or two 𝑡 to the first power, because we bring the exponent down to the front, the power down to the front, and then subtract one. And two minus one is one. So now for four 𝑡, our power would be one. The exponent would be one. We would bring that down to be with the four. And one times four would give us four. And then the power of 𝑡 was one. So we take one and subtract one, which would be zero. And 𝑡 to the zero power, actually anything to the zero power is simply one. And four times one is one. And then the derivative, the constant, is just zero. So we don’t have to do anything with the minus two.
So now let’s find d𝑦 d𝑡. So we take two 𝑡 squared. The power of two goes down in front to be with the two. And two times two is four. And then our exponent will turn into two minus one, so 𝑡 to the one power, which is just 𝑡. Now we have a constant. We have minus three. So its derivative is simply zero. So as we said before, to find d𝑦 d𝑥, we take d𝑦 d𝑡, which is four 𝑡, and divide it by d𝑥 d𝑡, two 𝑡 plus four.
Now we are able to take a factor of two out of the numerator and denominator, making d𝑦 d𝑥 equal to two 𝑡 over 𝑡 plus two. So now that we have d𝑦 d𝑥, we’re asked to find that at 𝑡 equals one. So if we would plug in one for 𝑡, we’ll have two times one over one plus two, resulting in two-thirds. Therefore, d𝑦 d𝑥 at 𝑡 equals one is equal to two over three or two-thirds.