Serlo: EN: Computing derivatives - special

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Special cases of the chain rule

Now we want to list a few special cases of the chain rule, which occur frequently in practice. For the derivation of the derivatives of $\exp$, $\ln$, $\sin$, $\cos$, $x\mapsto x^n$ etc. we refer to the following chapter Examples for derivatives (missing).

Case: $f$ is linear

Let $a,b\in ℝ$ and let $g:ℝ\to ℝ$ be differentiable. Then also $h:ℝ\to ℝ,h(x)=g(ax+b)$ is differentiable ad at $x\in ℝ$ there is

Case: $g$ is a power function

Let $f:D\to ℝ$ be differentiable. The also $f^n:D\to ℝ$ is differentiable for all $n\in \mathbb{N}$, where at $x\in D$ there is

Case: $g$ is a root function

Let $f:D\to {ℝ}^{+}$ be differentiable. then $\sqrt{f}:D\to {ℝ}^{+}$ with $x\mapsto \sqrt{f(x)}$ is differentiable as well and for all $x\in D$ there is

Case: $g=\exp$

Let $f:D\to ℝ$ be differentiable. Then $\exp \circ f:D\to ℝ$ is differentiable as well and for all $x\in D$ there is

Special case: Differentiating "function to the power of a function"

Consider the function

which is a special case of an exponential function. The inner function is $f=f_2\cdot (\ln \circ f_1)$. We may again just use the chain rule.

Case: $g=\ln$

Let $f:D\to ℝ\setminus \{0\}$ be and by the chain rule. Then, $\ln \circ |f|:D\to ℝ$ is and by the chain rule as well and for all $x\in D$ there is

Linear combinations of functions

The factor and sum rule state that the derivative is linear. If we apply this linearity to $n$ functions, we get:

Application: Deriving sum formulas

We can use the linearity of the derivative to obtain new sum formulas from already known ones. Let us consider as an example the geometric sum formula (missing) for $x\in ℝ\setminus \{1\}$ and $n\in \mathbb{N}$:

Both sides of the equation can be understood as differentiable functions $ℝ\setminus \{1\}$ or $f$ or $g$:

Since $f$ is a polynomial, we have for $x\in ℝ\setminus \{1\}$:

Furthermore, by the quotient rule

Since now $f\equiv g$, we also have $f^{\prime }\equiv g^{\prime }$. So for $x\in ℝ\setminus \{1\}$ there is:

Generalized product rule

The product rule $(f_1{f}_2{)}^{\prime }={f_1}^{\prime }{f}_2+f_1{f_2}^{\prime }$ can also be applied to more than two differentiable functions by first combining several functions and then applying the product rule several times in succession. For three functions we get

For four functions we get analogously

We now recognize a clear formation law for derivatives: the product of the functions is added up, whereby in each summand the derivative "moves forward" by one position. In general, the derivative of a product function of $n$ functions is:

Logarithmic derivatives

The logarithmic derivative is a very elegant tool to calculate the derivative of some functions of a special form. For a differentiable function $f$ without zeros, the logarithmic derivative is defined by

We have already shown above that the chain rule yields:

The following table lists some standard examples of logarithmic derivatives:

$f$	$\mathrm{L}(f)$	Domain of definition
$c\in ℝ\setminus \{0\}$	$\frac{0}{c}=0$	$ℝ$
$x^n$, $n\in \mathbb{N}$	$\frac{n{x}^{n-1}}{x^n}=\frac{n}{x}$	$ℝ\setminus \{0\}$
$\exp (x)$	$\frac{\exp (x)}{\exp (x)}=1$	$ℝ$
$\ln (x)$	$\frac{\frac{1}{x}}{\ln (x)}=\frac{1}{x\ln (x)}$	${ℝ}^{+}\setminus \{1\}$
$\sin (x)$	$\frac{\cos (x)}{\sin (x)}=\cot (x)$	$ℝ\setminus \{k\pi \mid k\in \mathbb{Z}\}$
$\cos (x)$	$\frac{-\sin (x)}{\cos (x)}=-\tan (x)$	$ℝ\setminus \{\frac{\pi }{2}+k\pi \mid k\in \mathbb{Z}\}$
$\tan (x)$	$\frac{\frac{1}{{\cos }^2(x)}}{\tan (x)}=\frac{1}{\sin (x)\cos (x)}$	$ℝ\setminus \{k\frac{\pi }{2}\mid k\in \mathbb{Z}\}$

By direct computation we obtain the following rules for the logarithmic derivative:

Using those rules, we can now easily calculate derivatives. The transition to logarithmic derivatives does not usually require less computational effort, but it is much clearer than calculating with the usual rules, and therefore less susceptible to errors!

Generalized chain rule

Just like the sum and product rule, the chain rule can be generalized to the composition of more than two functions. For two differentiable functions $f_1$ and $f_2$ the chain rule reads

If we have three functions $f_1$, $f_2$ and $f_3$, then by applying the rule twice we obtain

If we now take a closer look, we can see a law of formation: First the outermost function is differentiated and the two inner ones are inserted into the derivative function. Then the second function is differentiated and the innermost function is inserted, and the whole thing is multiplied by the first derivative. Finally, the innermost function is differentiated and multiplied. If we now generalize this to $n$ functions, we get:

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