3.3 Chain Rule

Example 1

Find the derivative of $y={(4x^3+15x)}^2$

This is not a simple polynomial, so we can’t use the basic building block rules yet. It is a product, so we could write it as $y={(4x^3+15x)}^2=(4x^3+15x)(4x^3+15x)$ and use the product rule. Or we could multiply it out and simply differentiate the resulting polynomial. I’ll do it the second way:
$$\begin{array}{rl}y=& {(4x^3+15x)}^2\\ =& 16x^6+120x^4+225x^2\\ y^{\prime }=& 96x^5+480x^3+450x\end{array}$$

Derivative Rules: Chain Rule

In what follows, $f$ and $g$ are differentiable functions with $y=f(u)$ and $u=g(x)$.

Chain Rule (Leibniz notation)

$$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$

Notice that the $du$’s seem to cancel. This is one advantage of the Leibniz notation – it can remind you of how the chain rule chains together.

Chain Rule (using prime notation)

$$\frac{d}{dx}\left[f(g(x))\right]=f^{\prime }(u)\cdot g^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)$$

Chain Rule (in words)

The derivative of a composition is the derivative of the outside (with the inside staying the same) TIMES the derivative of the inside.

Example 2

Find the derivative of $y={(4x^3+15x)}^2$

This is the same one we did before by multiplying out. This time, let’s use the Chain Rule: The inside function is what appears inside the parentheses: $4x^3+15x$. The outside function is the first thing we find as we come in from the outside – it’s the square function, $(\text{inside}{)}^2$.

The derivative of this outside function is $(2\cdot \text{inside})$. Now using the chain rule, the derivative of our original function is $(2\cdot \text{inside})$ TIMES the derivative of the inside (which is $12x^2+15$):
$$y^{\prime }=2(4x^3+15x)(12x^2+15)$$

Example 3

Find the derivative of $y={(4x^3+15x)}^{20}$.

Now we have a way to handle this one. It’s the derivative of the outside TIMES the derivative of the inside.

The outside function is ${\left(\text{inside}\right)}^{20}$, which has derivative $20{\left(\text{inside}\right)}^{19}$, so $$y^{\prime }=20{(4x^3+15x)}^{19}(12x^2+15).$$

Example 4

Differentiate $y=e^{x^2+5}$.

This isn’t a simple exponential function; it’s a composition. Typical calculator or computer syntax can help you see what the “inside” function is here. On a TI calculator, for example, when you push the $e^x$ key, it opens up parentheses: e^( . This tells you that the “inside” of the exponential function is the exponent. Here, the inside is the exponent $x^2+5$. Now we can use the Chain Rule: We want the derivative of the outside TIMES the derivative of the inside. The outside is the $e$ to the something function, so its derivative is the same thing. The derivative of what’s inside is $2x$. So $$\frac{d}{dx}\left(e^{x^2+5}\right)=\left(e^{x^2+5}\right)\cdot (2x).$$

Example 5

The table gives values for $f$ , $f^{\prime }$ , $g$, and $g^{\prime }$ at a number of points. Use these values to determine $(f\circ g)(x)$ and $(f\circ g{)}^{\prime }(x)$ at $x=-1$ and 0.

$(f\circ g)(-1)=f(g(-1))=f(3)=0$

$(f\circ g)(0)=f(g(0))=f(1)=1$

$(f\circ g{)}^{\prime }(-1)=f^{\prime }(g(-1))\cdot g^{\prime }(-1)=f^{\prime }(3)\cdot (0)=(2)(0)=0\text{ and}$

$(f\circ g{)}^{\prime }(0)=f^{\prime }(g(0))\cdot g^{\prime }(0)=f^{\prime }(1)\cdot (2)=(-1)(2)=-2$