Video: Differentiating Trigonometric Functions

Video Transcript

If 𝑦 equals seven tan two 𝑥, find 𝑑𝑦 𝑑𝑥.

In order to actually solve this problem and differentiate 𝑦 equals seven tan two 𝑥, we need to know a couple of sets of general rules. So the first one is that if 𝑦 is equal to tan 𝑥, then we know that 𝑑𝑦 𝑑𝑥 — so our derivative — is gonna be equal to sec squared 𝑥. However, if we take a look at our function, okay, yes, we’ve got 𝑦 is equal to tan 𝑥. But it’s not that simple because we’ve got 𝑦 is equal to seven tan two 𝑥.

So therefore, what we can actually say is that our function is more in this form 𝑦 equals 𝑎 tan 𝑓 𝑥. So what it’s saying is we’ve got a constant 𝑎 which is our seven and then tan of a function and our function is two 𝑥. Well, we can say that if it’s in this form, then 𝑑𝑦 𝑑𝑥 — so our derivative — is gonna be equal to 𝑎 multiplied by the derivative of our function 𝑓 𝑥 multiplied by sec squared 𝑓 𝑥 — so our function 𝑥.

Okay, so great, we now have this. Let’s use it to actually differentiate our function and find 𝑑𝑦 𝑑𝑥. Okay, so we have 𝑦 equals seven tan two 𝑥. Well, using our rule, we can actually say that 𝑑𝑦 𝑑𝑥 is gonna be equal to seven multiplied by the derivative of two 𝑥 multiplied by sec squared two 𝑥.

Well, if we differentiate two 𝑥 with respect to 𝑥, what we’re gonna get is two multiplied by one because it’s our coefficient multiplied by our exponent then 𝑥 to the power of one minus one which is gonna be equal to two 𝑥 to the power of zero. Well, we know that anything to the power of zero is one. So therefore, this is just gonna give us a result of two.

Okay, so let’s put this back in. So we’re gonna get that 𝑑𝑦 𝑑𝑥 is equal to seven multiplied by two multiplied by sec squared two 𝑥.

So therefore, we can say that if 𝑦 equals seven tan two 𝑥, then 𝑑𝑦 𝑑𝑥 is gonna be equal to 14 sec squared two 𝑥.