# LessonProduct Rule For n Factor Functions

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
This is a problem I enountered years ago and found interesting interesting:

• a) Determine $$\displaystyle f'(x)$$ given $$\displaystyle f(x)=\prod_{k=1}^n\left(g_k(x)\right)$$.
• b) If $$\displaystyle f(x)=\prod_{k=1}^n\left((1+kx)\right)$$ then find $$\displaystyle f'(0)$$
a) The product rule for a composite function that is the product of two functions is well-known and will be the basis for working this problem (and accepted without proof):

$$\displaystyle \frac{d}{dx}\left(g_1(x)\cdot g_2(x) \right)=g_1'(x)\cdot g_2(x)+g_1(x)\cdot g_2'(x)$$

Using this rule, let's look at:

$$\displaystyle \frac{d}{dx}\left(g_1(x)\cdot g_2(x)\cdot g_3(x) \right)$$

Now, let's associate two of the functions together, it doesn't matter which two, so let's use the first two:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)$$

Now, using the product rule above, we may state:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)=\frac{d}{dx}\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x)+\left(g_1(x)\cdot g_2(x) \right)\cdot g_3'(x)$$

Using the product rule again, we find:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)=\left(g_1'(x)\cdot g_2(x)+g_1(x)\cdot g_2'(x) \right)\cdot g_3(x)+\left(g_1(x)\cdot g_2(x) \right)\cdot g_3'(x)$$

And distributing, we find:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)=g_1'(x)\cdot g_2(x)\cdot g_3(x)+g_1(x)\cdot g_2'(x)\cdot g_3(x)+g_1(x)\cdot g_2(x)\cdot g_3'(x)$$

Now, this is enough to suggest the pattern (our induction hypothesis $$P_n$$):

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^n\left(g_k(x) \right) \right]=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^n\left(g_j(x) \right) \right]$$

Next, consider:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}(x) \right]$$

Using the product rule, and incorporating the new factor into the product. we may state:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\frac{d}{dx}\left[\prod_{k=1}^n\left(g_k(x) \right) \right]\cdot g_{n+1}(x)+\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}'(x)$$

Using our induction hypothesis, this becomes:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^n\left(g_j(x) \right) \right]\cdot g_{n+1}(x)+\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}'(x)$$

Now, incorporating the factor at the end of the first term on the right, we have:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^{n+1}\left(g_j(x) \right) \right]+\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}'(x)$$

And finally incorporating the second term on the right within the first summation term, we have:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\sum_{k=1}^{n+1}\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^{n+1}\left(g_j(x) \right) \right]$$

We have derived $$P_{n+1}$$ from $$P_n$$, thereby completing the proof by induction.

b) Now, if:

$$\displaystyle f(x)=\prod_{k=1}^n\left(g_k(x) \right)$$

and

$$\displaystyle g_k(x)=(1+kx)$$, we see that we have:

$$\displaystyle f'(x)=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(1+jx \right)\cdot k\cdot\prod_{j=k+1}^n\left(1+jx \right) \right]$$

Hence:

$$\displaystyle f'(0)=\sum_{k=1}^n(k)=\frac{n(n+1)}{2}$$

• anemone and harpazo

#### harpazo

##### Pure Mathematics
This is a problem I enountered years ago and found interesting interesting:

• a) Determine $$\displaystyle f'(x)$$ given $$\displaystyle f(x)=\prod_{k=1}^n\left(g_k(x)\right)$$.
• b) If $$\displaystyle f(x)=\prod_{k=1}^n\left((1+kx)\right)$$ then find $$\displaystyle f'(0)$$
a) The product rule for a composite function that is the product of two functions is well-known and will be the basis for working this problem (and accepted without proof):

$$\displaystyle \frac{d}{dx}\left(g_1(x)\cdot g_2(x) \right)=g_1'(x)\cdot g_2(x)+g_1(x)\cdot g_2'(x)$$

Using this rule, let's look at:

$$\displaystyle \frac{d}{dx}\left(g_1(x)\cdot g_2(x)\cdot g_3(x) \right)$$

Now, let's associate two of the functions together, it doesn't matter which two, so let's use the first two:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)$$

Now, using the product rule above, we may state:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)=\frac{d}{dx}\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x)+\left(g_1(x)\cdot g_2(x) \right)\cdot g_3'(x)$$

Using the product rule again, we find:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)=\left(g_1'(x)\cdot g_2(x)+g_1(x)\cdot g_2'(x) \right)\cdot g_3(x)+\left(g_1(x)\cdot g_2(x) \right)\cdot g_3'(x)$$

And distributing, we find:

$$\displaystyle \frac{d}{dx}\left(\left(g_1(x)\cdot g_2(x) \right)\cdot g_3(x) \right)=g_1'(x)\cdot g_2(x)\cdot g_3(x)+g_1(x)\cdot g_2'(x)\cdot g_3(x)+g_1(x)\cdot g_2(x)\cdot g_3'(x)$$

Now, this is enough to suggest the pattern (our induction hypothesis $$P_n$$):

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^n\left(g_k(x) \right) \right]=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^n\left(g_j(x) \right) \right]$$

Next, consider:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}(x) \right]$$

Using the product rule, and incorporating the new factor into the product. we may state:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\frac{d}{dx}\left[\prod_{k=1}^n\left(g_k(x) \right) \right]\cdot g_{n+1}(x)+\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}'(x)$$

Using our induction hypothesis, this becomes:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^n\left(g_j(x) \right) \right]\cdot g_{n+1}(x)+\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}'(x)$$

Now, incorporating the factor at the end of the first term on the right, we have:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^{n+1}\left(g_j(x) \right) \right]+\prod_{k=1}^n\left(g_k(x) \right)\cdot g_{n+1}'(x)$$

And finally incorporating the second term on the right within the first summation term, we have:

$$\displaystyle \frac{d}{dx}\left[\prod_{k=1}^{n+1}\left(g_k(x) \right) \right]=\sum_{k=1}^{n+1}\left[\prod_{j=1}^{k-1}\left(g_j(x) \right)\cdot\frac{d}{dx}\left(g_k(x) \right)\cdot\prod_{j=k+1}^{n+1}\left(g_j(x) \right) \right]$$

We have derived $$P_{n+1}$$ from $$P_n$$, thereby completing the proof by induction.

b) Now, if:

$$\displaystyle f(x)=\prod_{k=1}^n\left(g_k(x) \right)$$

and

$$\displaystyle g_k(x)=(1+kx)$$, we see that we have:

$$\displaystyle f'(x)=\sum_{k=1}^n\left[\prod_{j=1}^{k-1}\left(1+jx \right)\cdot k\cdot\prod_{j=k+1}^n\left(1+jx \right) \right]$$

Hence:

$$\displaystyle f'(0)=\sum_{k=1}^n(k)=\frac{n(n+1)}{2}$$
There are many problems like this in my calculus book by Larson and others that I would like to see full solutions to, honestly.

#### anemone

##### Paris la ville de l'amour
Staff member
Moderator
Math Helper
Awesome, Mark! Awesome!! You are really good at working with general case, just like part a of the problem! You are amazing to me!

• MarkFL

#### harpazo

##### Pure Mathematics
Awesome, Mark! Awesome!! You are really good at working with general case, just like part a of the problem! You are amazing to me!
Mark is an amazing mathematician.

• MarkFL