Differentiate the Law of Cosines with respect to time.

Step-by-step explanation:

The Law of Cosines is:

And we want to differentiate this with respect to time t.

So, let's take the derivative of both sides with respect to t:

First, let's expand our differentiation:

Implicitly differentiate. Let's ignore the third term for now aside from moving the two outside. Thus:

For the third term, this will require the product rule. We have:

The product rule states:

Therefore, we will let:

Then, we will take their derivatives with respect to t. Hence:

The left requires another application of the product rule. Let's ignore it for now. Implicitly differentiate the right:

On the left, we have:

Again, let's let: (The subscript is used to differentiate between variables)

Differentiate with respect to t:

Implicitly differentiate:

Hence, it follows that the above is:

Therefore:

Hence:

By substitution:

Distributive property:

Rearranging gives:

Factor out the derivative from the first and second brackets:

This cannot be simplified further.

Hence, the derivative of the Law of Cosines with respect to time t is:

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