The Chain Rule Basics | |

The Equation of the Tangent Line with the Chain Rule | |

More Practice |

The **chain rule** says when we’re taking the derivative, if there’s **something other than **$ \boldsymbol {x}$, like in the parenthesis or under a radical sign, for example, we have to **multiply what we get** **by the** **derivative of what’s inside the parentheses**. It all has to do with **Composite Functions**, since $ \displaystyle \frac{{dy}}{{dx}}=\frac{{dy}}{{du}}\cdot \frac{{du}}{{dx}}$. Note that we’ll learn how to “undo” the chain rule here in the **U-Substitution Integration** section.

Think of it this way when we’re thinking of rates of change, or derivatives: if we are running **twice** as fast as Person A, and then Person B is running **three times** as fast as us, Person B is running **six times** as fast as Person A. It’s all about relativity! Here is what it looks like in Theorem form:

If $ \displaystyle y=f\left( u \right)$ and $ u=g\left( x \right)$ are differentiable and $ y=f\left( {g\left( x \right)} \right)$, then:

$ \displaystyle \frac{{dy}}{{dx}}=\frac{{dy}}{{du}}\cdot \frac{{du}}{{dx}}$, or

$ \displaystyle \frac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right]={f}’\left( {g\left( x \right)} \right){g}’\left( x \right)$ (more simplified): $ \displaystyle \frac{d}{{dx}}\left[ {f\left( u \right)} \right]={f}’\left( u \right){u}’$

We’ve actually been using the chain rule all along, since the derivative of an expression with **just an **$ \boldsymbol {x}$** in it** is just **1**, so we are multiplying by **1**.

For example, if $ \displaystyle y={{x}^{2}},\,\,\,\,\,{y}’=2x\cdot \frac{{d\left( x \right)}}{{dx}}=2x\cdot 1=2x$.

Do you see how when we take the derivative of the “outside function” and there’s something **other than just $ \boldsymbol {x}$** in the **argument** (for example, in parentheses, under a radical sign, or in a trigonometric function), we have to take the derivative again of this “inside function”? In a nutshell, we are taking the derivative of the “outside function” and multiplying this by the derivative of the “inside” function(s). That’s pretty much it!

In the problems below, see how we take the derivative again of what’s in **red**? And sometimes, again, what’s in **blue**? Yes, sometimes we have to **use the chain rule twice**, in the cases where we have a function inside a function inside another function. We could theoretically take the chain rule a very large number of times, with one derivative!

Here’s one more problem, where we have to think about how the chain rule works:

## The Equation of the Tangent Line with the Chain Rule

Here are a few problems where we use the chain rule to find an **equation of the tangent line** to the graph $ f$ at the given point. Note that we saw more of these problems here in the **Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change** Section.

**Understand these problems, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

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**On to Implicit Differentiation and Related Rates – you’re ready!**