Derivative with Respect to a Variable
When we write $\dfrac{d}{dx}$, we explicitly say: differentiate with respect to $x$. This has two immediate consequences: $\dfrac{d}{dx}(x)=1$ and anything that does not depend on $x$ has derivative $0$.
📜 Core Facts
With respect to $x$:
$$ \frac{d}{dx}(x) = 1, \qquad \frac{d}{dx}(c) = 0 \ \text{for any constant } c. $$Variables that are independent of $x$ behave like constants:
$$ \frac{d}{dx}(y) = 0 \quad \text{if $y$ does not depend on $x$.} $$💡 Change of variable: If we differentiate with respect to $t$ instead, then $\dfrac{d}{dt}(t)=1$, and anything independent of $t$ has derivative $0$. 🧭 Multivariable note: In single-variable calculus, “independent of $x$” means “treated as constant”. In multivariable settings this is formalized by partial derivatives, e.g. $\partial/\partial x$.
🪜 How to Think About It (3 Steps)
- Fix the variable of differentiation: Decide whether you use $\dfrac{d}{dx}$, $\dfrac{d}{dt}$, etc.
- Treat other symbols as constants (if independent): Any symbol not depending on that variable differentiates to $0$.
- Apply the usual rules: Use constant rule, sum rule, product/quotient rule, chain rule as needed.
✏️ Examples for Illustration
Example 1: Basic identities
- $\dfrac{d}{dx}(x) = 1$
- $\dfrac{d}{dx}(5) = 0$
- $\dfrac{d}{dx}(y) = 0$ if $y$ does not depend on $x$
Example 2: Mixed symbols, differentiating with respect to $x$
Given: $f(x) = 3x^2 + ay$, where $a$ and $y$ are independent of $x$.
Derivative: $$ f'(x) = 6x + 0 = 6x. $$
Example 3: Change of the variable of differentiation
Given: $g(t) = bt^3 + x$, where $b$ and $x$ are independent of $t$.
$$ \frac{d}{dt}g(t) = 3bt^2 + 0 = 3bt^2. $$
Example 4: When a “symbol” actually depends on $x$
Let $y = y(x)$ be a function of $x$. Then $y$ is not independent of $x$:
$$ \frac{d}{dx}(y) = \frac{dy}{dx} \neq 0 \ \text{in general.} $$
Example 5: Product with an independent symbol
Given: $h(x) = k\cdot \sin x$, where $k$ is independent of $x$.
$$ h'(x) = k\cdot \cos x. $$