derivative

A function of a real variable $f(x)$ is differentiable at a point $a$ of its domain, if its domain contains an open interval containing $a$, and the limit

exists.
More explicitly, if the function $f$ is differentiable at $a$, that is if the limit $L$ exists, then this limit is called the derivative of $f$ at $a$.

If $f$ is a function that has a derivative at every point in its domain, then a function can be defined by mapping every point $x$ to the value of the derivative of $f$ at $x$; it is called the derivative function or the derivative of $f$, and we say $f$ is differentiable.

Let $f:x\mapsto y$ be a differentiable function such that $y=f(x)$.

The derivative of $f$ at $x$ is written

Derivative Rules

Let $f,g$ be differentiable functions and $a,b$ be constants.

• Constant Rule: $a^{\prime }=0$
• Sum Rule: $(af+bg{)}^{\prime }=a{f}^{\prime }+b{g}^{\prime }$
• Product Rule: $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$
• Quotient Rule: ${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$
• Chain Rule: $\frac{d}{dx}(f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Common Derivatives

• $\frac{d}{dx}{x}^a=a{x}^a-1$

• $\frac{d}{dx}{e}^x=e^x$

• $\frac{d}{dx}{a}^x=a^x\ln (a)$

• $\frac{d}{dx}\ln (x)=\frac{1}{x}$

• $\frac{d}{dx}{\log }_a(x)=\frac{1}{x\ln (a)}$
  n.b. the logarithm is only defined for inputs $>0$

• $\frac{d}{dx}\sin (x)=\cos (x)$

• $\frac{d}{dx}\cos (x)=-\sin (x)$

• $\frac{d}{dx}\tan (x)={\sec }^2(x)$

• $\frac{d}{dx}\sec (x)=\sec (x)\tan (x)$

• $\frac{d}{dx}\csc (x)=-\csc (x)\cot (x)$

• $\frac{d}{dx}\cot (x)=-{\csc }^2(x)$

• $\frac{d}{dx}\arcsin (x)=\frac{1}{\sqrt{1-x^2}}$

• $\frac{d}{dx}\arccos (x)=\frac{-1}{\sqrt{1-x^2}}$

• $\frac{d}{dx}\arctan (x)=\frac{1}{1+x^2}$

See also
antiderivative
Fundamental Theorem of Calculus