Operator Notation for Derivatives and PDEs

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Single Variable Differentiation

$$ \frac{d}{dx}x^n = nx^{n-1}\frac{d}{dx}x = nx^{n-1}\frac{dx}{dx} = nx^{n-1} \qquad [\text{chain/power rule}] $$

$$ \frac{d}{dx}y^n = ny^{n-1}\frac{d}{dx}y = ny^{n-1}\frac{dy}{dx} = ny^{n-1}y’ \qquad [\text{implicit differentiation}] $$

$$ \frac{d}{dx}y^n x^n = y^n[nx^{n-1}] + x^n[ny^{n-1}\color{red}{y’}] \qquad [\text{implicit}] $$

$$ \frac{d}{dt}xy = \frac{d}{dt}x(t)y(t) = x(t)\frac{d}{dt}y(t) + y(t)\frac{d}{dt}x(t) \qquad [\text{product rule}] $$

$$ \frac{d}{dt}ab = ab\frac{d}{dt}(1) = ab(0) = 0 \qquad [\text{constant rule}] $$

Multivariable Differentiation

$$ \frac{\partial}{\partial x}x^n = \frac{d}{dx}x^n = nx^{n-1} \qquad [\text{power rule}] $$

$$ \frac{\partial}{\partial x}y^n x^n = y^n\frac{\partial}{\partial x}x^n = y^n\frac{d}{dx}x^n = y^n[nx^{n-1}] \qquad [\text{constant multiple rule}] $$

Pay attention to the difference between $\frac{\partial}{\partial x}y^n x^n$ and $\frac{d}{dx}y^n x^n$ which is a constant multiple scenarios for partials vs. a product rule with implicit differential for single variable.

Solve a PDE (partial differential equation). Consider $u_x=x$.

$$ u_x = \frac{\partial u}{\partial x} = x \quad\Rightarrow\quad \partial u = x\,\partial x \quad\Rightarrow\quad u(x) = \frac{1}{2}x^2+g(y), $$

$$ \frac{\partial}{\partial x}g(y) = 0 \quad [\text{constant function}], \qquad \frac{d}{dx}C = 0 \quad [\text{constant}]. $$

Going backwards

$$ \frac{\partial}{\partial x} \left[ u(x)=\frac{1}{2}x^2+g(y) \right] \quad\Rightarrow\quad \frac{\partial u}{\partial x} = \frac{1}{2}[2x^{2-1}] + \frac{\partial}{\partial x}g(y) \quad\Rightarrow\quad u_x=x. $$

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