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.
