Differentiation

One of the main branches of calculus, counterpart to integration. The core idea is that we measure rate of change.

Interpretation

For some function $f$:

• $f^{\prime }(x)>0$, then $f$ is increasing
• $f^{\prime }(x)<0$, then $f$ is decreasing
• $f^{\prime \prime }(x)>0$, then $f$ is concave up
• $f^{\prime \prime }(x)<0$, then $f$ is concave down

Key concepts

Foundations

• Differentiation from first principles
• Differentiability
• Implicit differentiation
• Partial differentiation
• Fundamental theorem of calculus
• Chain rule

Applications

• Inverse differentiation
  • Related rates
• Kinematics
• L’Hopital’s rule
• Differentiator

Computer methods

The Symbolic Math Toolbox has a differentiation command, which works with functions of multiple variables:

syms x;
y = x^2 + x;
diff(y, x) % second input is variable to take derivative wrt

Useful rules

For $y=uv$, we use the product rule:

$$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$$

For $y=\frac{u}{v}$, we use the quotient rule:

$$\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$$

For some $y=g(u)$ and $u=f(x$) we use the chain rule:

$$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$

Standard derivatives

$$\frac{d}{dx}{x}^n=n{x}^{n-1}$$ $$\frac{d}{dx}\sin x=\cos x$$ $$\frac{d}{dx}\cos x=-\sin x$$ $$\frac{d}{dx}{e}^x=e^x$$ $$\frac{d}{dx}\ln x=\frac{1}{x}$$ $$\frac{d}{dx}\tan x={\sec }^{2}x$$ $$\frac{d}{dx}\csc x=-\csc x\cot x$$ $$\frac{d}{dx}\sec x=\tan x\sec x$$ $$\frac{d}{dx}\cot x=-{\csc }^{2}x$$ $$\frac{d}{dx}{a}^x=a^x(\ln a)$$ $$\frac{d}{dx}{\log }_{a}x=\frac{1}{x\ln a}$$

See inverse differentiation for a derivation of the derivatives of inverse trigonometric functions.

$$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$$ $$\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}$$ $$\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$$