Find the derivative of ln(1 + x/(e*pi)) with respect to x.

Step-by-step solution

1. 1

   Treat $e\pi$ as a single constant. The product $e\pi\approx 8.5397$ contains no $x$, so it behaves exactly like any other number. Recognising this early prevents the temptation to apply a product or quotient rule to it.

2. 2

   Apply the chain rule for the natural logarithm. With $u=1+\dfrac{x}{e\pi}$,

   $\frac{d}{dx}\ln u=\frac{u'}{u},\qquad u'=\frac{1}{e\pi}$

   The inner derivative is the constant $\tfrac{1}{e\pi}$ because $x$ appears linearly.

3. 3

   Write the quotient.

   $\frac{dy}{dx}=\frac{\frac{1}{e\pi}}{1+\frac{x}{e\pi}}$

   This is already correct but is a fraction inside a fraction; simplifying is worth one more line.

4. 4

   Clear the nested fraction. Multiply numerator and denominator by $e\pi$:

   $\frac{\frac{1}{e\pi}\cdot e\pi}{\left(1+\frac{x}{e\pi}\right)e\pi}=\frac{1}{e\pi+x}$

   The constant disappears from the numerator entirely and simply shifts the denominator.

5. 5

   Recognise the general pattern and check. For any constant $c$, $\dfrac{d}{dx}\ln\left(1+\tfrac{x}{c}\right)=\dfrac{1}{x+c}$: the derivative of the log of a linear function is one over that linear function, scaled so the leading coefficient is 1. Numerically at $x=1.7$ the central difference is $0.0976588$ and $\dfrac{1}{1.7+8.5397}=0.0976588$ ✓.