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

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Lesson Parent: 
Chapter 11 - The Chain Rule
Problem: 

Determine $(g\circ f)\,'(x)$ 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 $\frac{dg}{dx}=4$
Place $f$ in for every value of $x$ in $\frac{dg}{dx}$. Call this $\frac{dg}{df}$. \[\frac{dg}{df}=4\]
Place the function $f(x)$ into every value of $f$ in $\frac{dg}{df}$. We still call this $\frac{dg}{df}$. \[\frac{dg}{df}=4\]
Multiply $\frac{df}{dx}$ times $\frac{dg}{df}$. This is $\frac{d}{dx}\left[g\circ f\right]$. \[\frac{dg}{df}\cdot\frac{df}{dx}=4\cdot \left[6x+5\right]\]
If needed, now do some algebra to simplify the problem. \[\frac{dg}{df}\cdot\frac{df}{dx}=24x+20\]