Differentiation of General Functions

1. Concept Overview

Differentiation tells us how fast a function changes with respect to its variable.
If [y = f(x)], then the derivative is written as:

[f'(x)] or [\dfrac{dy}{dx}]

It represents the rate of change or slope of the tangent to the curve at a point.

2. Standard Derivatives — MUST MEMORIZE

Algebraic + Exponential + Logarithmic + Trigonometric

Function [f(x)]	Derivative Symbol	Result
[x^n]	[\dfrac{d}{dx}(x^n)]	[nx^{,n-1}]
[e^x]	[\dfrac{d}{dx}(e^x)]	[e^x]
[a^x]	[\dfrac{d}{dx}(a^x)]	[a^x \ln(a)]
[\ln(x)]	[\dfrac{d}{dx}(\ln(x))]	[\dfrac{1}{x}]
[\log_a(x)]	[\dfrac{d}{dx}(\log_a(x))]	[\dfrac{1}{x \ln(a)}]
[\sin(x)]	[\dfrac{d}{dx}(\sin(x))]	[\cos(x)]
[\cos(x)]	[\dfrac{d}{dx}(\cos(x))]	[-\sin(x)]
[\tan(x)]	[\dfrac{d}{dx}(\tan(x))]	[\sec^2(x)]
[\cot(x)]	[\dfrac{d}{dx}(\cot(x))]	[-\csc^2(x)]
[\sec(x)]	[\dfrac{d}{dx}(\sec(x))]	[\sec(x)\tan(x)]
[\csc(x)]	[\dfrac{d}{dx}(\csc(x))]	[-\csc(x)\cot(x)]

3. Inverse Trigonometric Functions

Function [f(x)]	Derivative Symbol	Result
[\sin^{-1}(x)]	[\dfrac{d}{dx}(\sin^{-1}(x))]	[\dfrac{1}{\sqrt{1-x^2}}]
[\cos^{-1}(x)]	[\dfrac{d}{dx}(\cos^{-1}(x))]	[-\dfrac{1}{\sqrt{1-x^2}}]
[\tan^{-1}(x)]	[\dfrac{d}{dx}(\tan^{-1}(x))]	[\dfrac{1}{1+x^2}]
[\cot^{-1}(x)]	[\dfrac{d}{dx}(\cot^{-1}(x))]	[-\dfrac{1}{1+x^2}]
[\sec^{-1}(x)]	[\dfrac{d}{dx}(\sec^{-1}(x))]	$ \displaystyle \left\{ \begin{array}{l}\dfrac{1}{{x\sqrt{{{{x}^{2}}-1}}}},\text{ }x>1\\-\dfrac{1}{{x\sqrt{{{{x}^{2}}-1}}}},\text{ }x<-1\end{array} \right.$
[\csc^{-1}(x)]	[\dfrac{d}{dx}(\csc^{-1}(x))]	$ \displaystyle \left\{ \begin{array}{l}-\dfrac{1}{{x\sqrt{{{{x}^{2}}-1}}}},\text{ }x>1\\\dfrac{1}{{x\sqrt{{{{x}^{2}}-1}}}},\text{ }x<-1\end{array} \right.$

4. Key Features

✔ Differentiation works term-by-term
✔ No need to expand each time—use rules
✔ Helps calculate slopes, velocity, marginal change, etc.
✔ Essential foundation for further calculus topics

Unlock the full course today

Get full access to all videos and content.