Video: Differentiating Trigonometric Functions

If 𝑦 = 7 tan 2π‘₯, find 𝑑𝑦/𝑑π‘₯.

02:00

Video Transcript

If 𝑦 equals seven tan two π‘₯, find 𝑑𝑦 𝑑π‘₯.

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

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

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

Well, if we differentiate two π‘₯ with respect to π‘₯, what we’re gonna get is two multiplied by one because it’s our coefficient multiplied by our exponent then π‘₯ to the power of one minus one which is gonna be equal to two π‘₯ 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 𝑑π‘₯ is equal to seven multiplied by two multiplied by sec squared two π‘₯.

So therefore, we can say that if 𝑦 equals seven tan two π‘₯, then 𝑑𝑦 𝑑π‘₯ is gonna be equal to 14 sec squared two π‘₯.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.