3.3 - Basic Derivative Rules

#calc/concept

Derivative of a Constant Function

If $f$ is the function with constant value $c$, then ${\displaystyle \frac{df}{dx}=\frac{d}{dx}(c)=0}$.

• Essentially, derivative of a constant is always $0$.

Power rule for Positive and Negative Integer Powers of $x$

If $n$ is a positive integer, then ${\displaystyle \frac{d}{dx}{x}^n=n{x}^{n-1}}$. This works for negative integer powers too.

Constant Multiple Rule

If $u$ is a differentiable function of $x$ and $c$ is a constant, then ${\displaystyle \frac{d}{dx}(cu)=c\frac{du}{dx}}$.

• So derivative of $3x^4$ is $3$ times the derivative of $x^4$.

Sum and Difference Rule

If $u$ and $v$ are differentiable functions of $x$, then their sum and difference are differentiable at every point where $u$ and $v$ are differentiable. At such points, ${\displaystyle \frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}}$

• Basically, you can separately take derivatives of different terms of the polynomial.

Product Rule

The product of two differentiable functions $u$ and $v$ is differentiable, and ${\displaystyle \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}}$.

• In words, (first function)(derivative of second function) $+$ (second function)(derivative of first function).

Quotient Rule

At a point where $v\ne 0$, the quotient $y=\frac{u}{v}$ of two differentiable functions is differentiable, and ${\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}$.

• If it helps, use the mnemonic “Low D High - High D Low”.