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Using the product rule and the chain rule - Video Lecture - Engineering
FAQs on Using the product rule and the chain rule - Mathematics
| 1. What is the product rule in mathematics? |  |
The product rule is a rule in calculus that allows us to differentiate the product of two functions. It states that if we have two functions, f(x) and g(x), the derivative of their product, h(x) = f(x) * g(x), is given by h'(x) = f'(x) * g(x) + f(x) * g'(x).
| 2. How do we apply the product rule to differentiate a function? | |
To apply the product rule, we need to differentiate each function separately and then combine the results. Let's say we have a function h(x) = f(x) * g(x). We differentiate f(x) to get f'(x) and differentiate g(x) to get g'(x). Then, we use the product rule formula: h'(x) = f'(x) * g(x) + f(x) * g'(x). This gives us the derivative of the product of two functions.
| 3. What is the chain rule in calculus? | |
The chain rule is a rule in calculus that allows us to differentiate composite functions. A composite function is a function that is formed by applying one function to the output of another function. The chain rule states that if we have a composite function y = f(g(x)), the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).
| 4. How do we apply the chain rule to differentiate a composite function? | |
To apply the chain rule, we first identify the outer function and the inner function of the composite function. Let's say we have y = f(g(x)). We differentiate the outer function f(g(x)) with respect to its inner function g(x), which gives us f'(g(x)). Then, we differentiate the inner function g(x) with respect to x, which gives us g'(x). Finally, we multiply these two derivatives together to obtain the derivative of the composite function: dy/dx = f'(g(x)) * g'(x).
| 5. Can we use both the product rule and the chain rule together? | |
Yes, we can use both the product rule and the chain rule together when differentiating a function that involves both products and composite functions. If a function contains both a product of functions and a composite function, we first apply the product rule to differentiate the product part, and then use the chain rule to differentiate the composite part. This allows us to find the derivative of complex functions that involve both products and compositions.
Text Transcript from Video
Let's now use what we know about the chain
rule and
the product rule to take the derivative of
an even weirder expression.
So, we're gonna take the derivative, we're
gonna take the derivative of
either the cosine of x times the cosine of
e to the x.
So, let's take the derivative of this.
So, we can view this as the product of two
functions.
So, the product rule tells us that this is
going to be
the derivative with respect to x of e to
the cosine of x,
e to the cosine of x times cosine times
cosine of e to the x
plus, plus the first function, just e to
the cosine of x.
E to the cosine of x, times the derivative
of the second function.
Times the derivative with respect to x of
cosine of e to the x.
Cosine of e to the x.
And so, we just need to figure out what
these two derivatives are.
And so you can imagine the chain rule
might be applicable here.
So, let me make it clear.
This, we got from the product rule.
Product, product rule.
But then, to evaluate each of these
derivatives, we need to use the chain
rule.
So, let's think about this a little bit.
So, the derivative, let me copy and paste
this so I don't have to rewrite it.
So, copy and paste.
So, let's think about what the derivative
of e to
the cosine of x is, e to the cosine of x.
So, we can view our outer function as e to
the something
,as e to the something and the derivative
of e to the
something with respect to something is
just going to be e to
that something so it's going to be e to
the cosine of x.
So, let me do that in that same blue
color.
So, it's going to be e, actually I'm gonna
do it
in that, actually I'm gonna do it in a new
color.
Let me do it in magenta.
So, the derivative of e to the something
with
respect to something is just e to the
something.
It's just e to the cosine of x, and we
have to
multiply that times the derivative of the
something with respect to x.
So, what's the derivative of cosine of x
with respect to x?
Well, that's just negative sine of x.
So, it's times negative sine of x.
And so, we figured out this first
derivative.
Let me make it clear.
This right over here is the derivative of
e to the cosine of x.
Derivative of e to the cosine of x, with
respect to, with respect to cosine of x.
And this right over here, this right over
here is the
derivative of cosine of x with respect,
with respect to x.
And we just took the product of the two.
That's what the chain rule tells us.
Fair enough.
Now, let's figure out this derivative out
here.
So, we want to find the derivative with
respect to x of cosine e to the x.
So, once again, let me copy and paste it.
So, we need to figure out this thing right
over here.
So first, just like we did, we're just
going to apply the chain rule again.
We need to figure out the derivative of
cosine of something,
in this case, e to the x, with respect to
that something.
So, this is going to be equal to
derivative of cosine of something
with respect to that something is equal to
the negative sine of that something.
Negative sine of e to x, of e to the x.
Once again, we can view this as the
derivative of cosine
of e to the x, with respect to, with
respect to e to the x.
And then we multiply that times the
derivative of the something with respect
to x.
So, let me do this, and this, and I'm
running out of colors.
Let me do this in the screen color.
So, times the derivative of e to the x
with respect to x is just e to the x.
So that right over there is the derivative
of
e to the x with respect, with respect to
x.
And so, we're essentially done, we just
have to substitute what
we found using the chain rule back into
our original expression.
The derivative of this business up here is
going to be equal to, let
me just copy and paste everything just to
make everything nice, nice and clean.
So, copy and paste.
So, that is going to be equal to, is going
to be equal to
this times cosine e to the x, so this is
going to be, we'll see,
we could put the e to the x out front, we
could put the
negative out front, so we could write it
as negative e to the cosine x.
e to the Cosine x, times Sine of x, times
Sine of x,
times Cosine of e to the x, times Cosine
of e to the x.
So, that's this first term here.
Plus, plus e of the cosine x times all of
this stuff.
And so, let's see, we can put the negative
out front again.
So let's put that negative out front.
So we have a negative, negative of e to
the cosine of x times e to the x.
I can write it this way.
e to the x times e to the cosine of x, and
you could simplify that, or combine
it, since you're multiplying two things
with the same
base, but I'll just leave it like this,
like this.
e to the x, times e to the cosine of x,
times the sine.
We already have the negative, so then we
have sine of e to the x.
Sine of e to the x, right over here.
Times sin, sin, of e to the x, we had
negative sin e
to the x times e to the x, negative sin of
e to the
x times e to the x, and then that
multiplied by e to
the cosine of x so we have the exact same
thing right over here.
And we're done.
rule and
the product rule to take the derivative of
an even weirder expression.
So, we're gonna take the derivative, we're
gonna take the derivative of
either the cosine of x times the cosine of
e to the x.
So, let's take the derivative of this.
So, we can view this as the product of two
functions.
So, the product rule tells us that this is
going to be
the derivative with respect to x of e to
the cosine of x,
e to the cosine of x times cosine times
cosine of e to the x
plus, plus the first function, just e to
the cosine of x.
E to the cosine of x, times the derivative
of the second function.
Times the derivative with respect to x of
cosine of e to the x.
Cosine of e to the x.
And so, we just need to figure out what
these two derivatives are.
And so you can imagine the chain rule
might be applicable here.
So, let me make it clear.
This, we got from the product rule.
Product, product rule.
But then, to evaluate each of these
derivatives, we need to use the chain
rule.
So, let's think about this a little bit.
So, the derivative, let me copy and paste
this so I don't have to rewrite it.
So, copy and paste.
So, let's think about what the derivative
of e to
the cosine of x is, e to the cosine of x.
So, we can view our outer function as e to
the something
,as e to the something and the derivative
of e to the
something with respect to something is
just going to be e to
that something so it's going to be e to
the cosine of x.
So, let me do that in that same blue
color.
So, it's going to be e, actually I'm gonna
do it
in that, actually I'm gonna do it in a new
color.
Let me do it in magenta.
So, the derivative of e to the something
with
respect to something is just e to the
something.
It's just e to the cosine of x, and we
have to
multiply that times the derivative of the
something with respect to x.
So, what's the derivative of cosine of x
with respect to x?
Well, that's just negative sine of x.
So, it's times negative sine of x.
And so, we figured out this first
derivative.
Let me make it clear.
This right over here is the derivative of
e to the cosine of x.
Derivative of e to the cosine of x, with
respect to, with respect to cosine of x.
And this right over here, this right over
here is the
derivative of cosine of x with respect,
with respect to x.
And we just took the product of the two.
That's what the chain rule tells us.
Fair enough.
Now, let's figure out this derivative out
here.
So, we want to find the derivative with
respect to x of cosine e to the x.
So, once again, let me copy and paste it.
So, we need to figure out this thing right
over here.
So first, just like we did, we're just
going to apply the chain rule again.
We need to figure out the derivative of
cosine of something,
in this case, e to the x, with respect to
that something.
So, this is going to be equal to
derivative of cosine of something
with respect to that something is equal to
the negative sine of that something.
Negative sine of e to x, of e to the x.
Once again, we can view this as the
derivative of cosine
of e to the x, with respect to, with
respect to e to the x.
And then we multiply that times the
derivative of the something with respect
to x.
So, let me do this, and this, and I'm
running out of colors.
Let me do this in the screen color.
So, times the derivative of e to the x
with respect to x is just e to the x.
So that right over there is the derivative
of
e to the x with respect, with respect to
x.
And so, we're essentially done, we just
have to substitute what
we found using the chain rule back into
our original expression.
The derivative of this business up here is
going to be equal to, let
me just copy and paste everything just to
make everything nice, nice and clean.
So, copy and paste.
So, that is going to be equal to, is going
to be equal to
this times cosine e to the x, so this is
going to be, we'll see,
we could put the e to the x out front, we
could put the
negative out front, so we could write it
as negative e to the cosine x.
e to the Cosine x, times Sine of x, times
Sine of x,
times Cosine of e to the x, times Cosine
of e to the x.
So, that's this first term here.
Plus, plus e of the cosine x times all of
this stuff.
And so, let's see, we can put the negative
out front again.
So let's put that negative out front.
So we have a negative, negative of e to
the cosine of x times e to the x.
I can write it this way.
e to the x times e to the cosine of x, and
you could simplify that, or combine
it, since you're multiplying two things
with the same
base, but I'll just leave it like this,
like this.
e to the x, times e to the cosine of x,
times the sine.
We already have the negative, so then we
have sine of e to the x.
Sine of e to the x, right over here.
Times sin, sin, of e to the x, we had
negative sin e
to the x times e to the x, negative sin of
e to the
x times e to the x, and then that
multiplied by e to
the cosine of x so we have the exact same
thing right over here.
And we're done.
About this Video
Apr 19, 2026 Last updated
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