How to Find the Derivative

Rules of Differentiation

For the functions that involve operations on other functions, there are six rules of differentiation: Coefficient Rule, Addition Rule, Subtraction Rule, Product Rule, Quotient Rule, and Chain Rule.

The Coefficient Rule states that for a constant $k\in \mathbb{R}$ and the function $f(x)$

${(k\cdot f(x))}^{\prime }=k\cdot f^{\prime }(x)$

Example

${(3\cdot \sin x)}^{\prime }=3\cdot {(\sin x)}^{\prime }=\boxed{{\displaystyle 3\cos x}}$

The Addition Rule states that for the functions $f(x)$ and $g(x)$

${(f(x)+g(x))}^{\prime }=f^{\prime }(x)+g^{\prime }(x)$

Example

${(x^2+e^x)}^{\prime }={\left(x^2\right)}^{\prime }+{\left(e^x\right)}^{\prime }=\boxed{{\displaystyle 2x+e^x}}$

The Subtraction Rule states that for the functions $f(x)$ and $g(x)$

${(f(x)-g(x))}^{\prime }=f^{\prime }(x)–g^{\prime }(x)$

Example

${(x^3–\ln x)}^{\prime }={\left(x^3\right)}^{\prime }–{(\ln x)}^{\prime }=\boxed{{\displaystyle 3x^2-\frac{1}{x}}}$

The Product Rule states that for the functions $f(x)$ and $g(x)$

${(f(x)\cdot g(x))}^{\prime }=f^{\prime }(x)\cdot g(x)+g^{\prime }(x)\cdot f(x)$

${(x\cdot \tan x)}^{\prime }={\left(x\right)}^{\prime }\cdot (\tan x)+{(\tan x)}^{\prime }\cdot \left(x\right)=\left(1\right)\cdot (\tan x)+({\sec }^{2}x)\cdot \left(x\right)=\boxed{{\displaystyle \tan x+x{\sec }^{2}x}}$

The Quotient Rule states that for the functions $f(x)$ and $g(x)$

${\left(\frac{f(x)}{g(x)}\right)}^{\prime }=\frac{f^{\prime }(x)\cdot g(x)–g^{\prime }(x)\cdot f(x)}{g^2(x)}$

Example