**What is the Chain Rule? **

**It is a mathematical formula for calculating the derivative of composite functions such as f (g(x))**

** **

**If we have two functions:**

*Y = f (u) , u = g (x)*

**[ The two preceding functions can also be expressed as f (g(x)) ]**

**Then **

**[The derivative of "y" function with respect to x (dy/ dx) = The derivative of "y" function with respect to u (dy/du) Multiplied by the derivative of "u" function with respect to x (du/dx) ]**

**Steps to find the first derivative of a composite function using the ***Chain Rule*:-

*Chain Rule*:-

**Find****y ′****[The derivative of y function with respect to u (dy/du) ].****Find****u ′****[The derivative of u function with respect to x (du/dx) ].****Multiply the two derivatives.****Substitute for the value of***u*.**Substitute for the value of***x*-if any.

**Example**

**Use the Chain Rule to find the first derivative of the following function when**

*x*=3:-**Solution:-**

**1- We rewrite the function y with respect to u (we form two functions. The first function is the function "**

*y*" with respect to u, and the second function is the function "*u*" with respect to x). In this problem we will impose what is under the square root with u.**2- Find**

**y ′**

**[The derivative of "**

*y*" function with respect to u (dy/du) ]**3- Find**

**u ′****[The derivative of "**

*u*" function with respect to x (du/dx) ]**4- Multiply the two derivatives**

**5- Substitute for the value of**

*u***6- Substitute for the value of**

*x*