# Does it make sense to multiply slopes?

Multiplying fractions is a regular occurance. If those fractions are considered slopes, does it make any sense?

For example, if these fractions are slopes,$\frac{9}{8} \times \frac{49}{48},$ does the product have meaning in terms of slope?

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## 1 Answer

Sure. Let $f(x)=ax+b, g(x)=cx+d$. Then the slope of $f(g(x))$ is $ac$.

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Thanks, What would one be trying to show if one did this? In the example above, two rising slopes become one steeper rising slope. Does that have meaning anywhere? –  Fred Kline Apr 18 at 14:08
The product of two rising (i.e., positive) slopes isn't necessarily steeper: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. –  Joshua Taylor Apr 18 at 15:26
@FredKline If you remove the translation factor (so that you're dealing with functions whose graphs are lines through the origin), then this becomes a special case of what happens when you compose linear transformations. The phenomenon is that composing linear transformations is the same thing as multiplying numerical data describing those transformations in the appropriate way. The numerical data are the matrices corresponding to the transforms, and the correct notion of multiplication is matrix multiplication. In other words, what you've hit on runs deeper than one might originally expect. –  Nick Apr 18 at 18:29
This is related with the chain rule for differentation as well :) –  Mon Kee Poo Apr 18 at 18:52