Determine $(f\circ g)\,'(x)$ where $f(x)=3x^2+5x-2$ and $g(x)=4x+3$.

Primary tabs

Lesson Parent: 
Chapter 11 - The Chain Rule
Problem: 

Determine $(f\circ g)\,'(x)$ where $f(x)=3x^2+5x-2$ and $g(x)=4x+3$.

Solution: 

Let's follow our steps given here:

Step Example
Determine $\frac{df}{dx}$ and $\frac{dg}{dx}$. (Recall the derivative of a polynomial here.) $\frac{df}{dx}=6x+5$ and and $\frac{dg}{dx}=4$
Place $g$ in for every value of $x$ in $\frac{df}{dx}$. Call this $\frac{df}{dg}$. \[\frac{df}{dg}=6g+5\]
Place the function $g(x)$ into every value of $g$ in $\frac{df}{dg}$. We still call this $\frac{df}{dg}$. \[\frac{df}{dg}=6(4x+3)+5\]
Multiply $\frac{dg}{dx}$ times $\frac{df}{dg}$. This is $\frac{d}{dx}\left[f\circ g\right]$. \[\frac{df}{dg}\cdot\frac{dg}{dx}=\left[6(4x+3)+5\right]\cdot 4\]
If needed, now do some algebra to simplify the problem. \[\frac{df}{dg}\cdot\frac{dg}{dx}=96x+92\]