Q.Differentiate w.r.t. $x$: $\dfrac{8^x}{x^8}$.

The key idea is to use logarithmic differentiation (or rewrite $8^x = e^{x \log 8}$) because the variable appears in both the base and the exponent.

Let $y = \dfrac{8^x}{x^8}$. Taking natural logs on both sides:

Differentiate implicitly with respect to $x$:

$\frac{1}{y} \frac{dy}{dx} = \log 8 - \frac{8}{x}$ …

The problem asks us to differentiate $\frac{8^x}{x^8}$ with respect to $x$. At first glance, this looks like a straightforward quotient rule problem. But there’s a subtlety: the numerator $8^x$ is an exponential function (base constant, exponent variable), while the denominator $x^8$ is a power function (base variable, exponent constant). Their derivatives are different in form, and mixing them in a quotient requires care.

The key idea is to apply the quotient rule:

where $u = 8^x$ and $v = x^8$. Then we need $u'$ and $v'$ correctly.

Let’s proceed step by step.

1. Identify $u$ and $v$

   Let $u = 8^x$ and $v = x^8$.

2. Differentiate $u = 8^x$

   The derivative of an exponential $a^x$ is $a^x \log a$. So:

No exponent reduction — just multiply by the natural log of the base.

3. Differentiate $v = x^8$ This is a standard power rule: bring down the exponent 8, reduce the exponent by 1:

4. Apply the quotient rule

5. Simplify the numerator Factor out the common term $8^x$: $\text{Numerator} = 8^x \left( x^8 \log 8 - 8x^7 \right).$ …

Method: Logarithmic Differentiation for Mixed Exponential-Power Expressions

Use logarithmic differentiation whenever an expression mixes a variable exponent (like $8^x$) with a variable base raised to a constant power (like $x^8$), especially in a product or quotient — direct product/quotient rules would need two different derivative formulas glued together, and it is easy to mix them up.

Steps

Step 1: Recognise when logarithmic differentiation helps

Whenever the expression is a product or quotient of terms with variables in both the base and the exponent (such as $\dfrac{8^x}{x^8}$), taking logarithms first turns a messy product/quotient into a simple sum/difference, which is far easier to differentiate correctly.

Step 2: Take the natural logarithm of both sides

For $y = \dfrac{8^x}{x^8}$:

using $\log(m/n)=\log m - \log n$ and $\log(m^k)=k\log m$.

Step 3: Differentiate implicitly with respect to $x$ …

Common Mistakes

Mistake 1: Applying the power rule to $8^x$ instead of the exponential rule

Why it's wrong: writing the derivative of $8^x$ as $8x^{7}$ (as if it were $x^8$) confuses a power function with an exponential function — the power rule only applies when the variable is in the base, not the exponent. The correct derivative of $8^x$ is $8^x \log 8$. Correct approach: always check whether the variable is in the base or the exponent before choosing power rule vs. exponential rule.

Mistake 2: Stopping at $\dfrac{y'}{y}=\log 8 - \dfrac{8}{x}$ and forgetting to multiply back by $y$ …