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## Exponential functions differentiation

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# Worked example: Derivative of 7^(x²-x) using the chain rule

AP.CALC:

FUN‑3 (EU)

, FUN‑3.C (LO)

, FUN‑3.C.1 (EK)

## Video transcript

- [Voiceover] Let's say
that y is equal to seven to the x squared minus x power. What is the derivative
of y, derivative of y, with respect to x? And like always, pause this video and see if you can figure it out. Well, based on how this has
been color-coded ahead of time, you might immediately recognize that this is a composite function,
or it could be viewed as a composite function. If you had a v of x, which
if you had a function v of x, which is equal to seven to the xth power, and you had another function u of x, u of x which is equal
to x squared minus x, then what we have right over here, y, y is equal to seven to something, so it's equal to v of,
and it's not just v of x, it's v of u of x, instead of an x here you have the whole function u of x, x squared minus x. So, it's v of u of x and
the chain rule tells us that the derivative of
y with respect to x, and you'll see different notations here, sometimes you'll see it
written as the derivative of v with respect to
u, so v prime of u of x times the derivative
of u with respect to x, so that's one way you could do it, or you could say that this is equal to, this is equal to the
derivative, the derivative of v with respect to x, sorry,
derivative of v with respect to u, d v d u times the derivative
of u with respect to x, derivative of u with respect to x, and so either way we can
apply that right over here. So, what's the derivative
of v with respect to u? What is v prime of u of x? Well, we know, we know,
let me actually write it right over here, if v of x is
equal to seven to the x power v prime of x would be equal
to, and we've proved this in other videos where we
take derivatives exponentials of bases other than e, this
going to be the natural log of seven times seven to the x power. So, if we are taking v prime of u of x, then notice instead of an x everywhere, we're going to have a u of x everywhere. So, this right over
here, this is going to be natural log of seven times seven to the, instead of saying seven to the x power, remember we're taking v prime of u of x, so it's going to be seven to
the x squared minus x power, x squared, x squared minus x power, and then we want to multiply
that times the derivative of u with respect to x. So, u prime of x, well, that's going to be two x to the first which
is just two x minus one, so we're going to
multiply this times two x, two x minus one, so there you have it, that is the derivative
of y with respect to x. You could, we could try to simplify this or I guess re-express
it in different ways, but the main thing to realize is, look, we're just gonna take the
derivative of the seven to the this to the u of x
power with respect to u of x. So, we treat the u of x the
way that we would've treated an x right over here, so it's
gonna be natural log of seven times seven to the u of x power, we take that and multiply
that times u prime of x, and once again this is just an application of the chain rule.