📐 Taking Derivatives
What Is a Derivative?
The derivative of a function $f(x)$ measures how $f$ changes with respect to $x$—its instantaneous rate of change. Formally,
$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. $$It underpins linear approximations, optimization with gradient methods, and the geometry of curves.
Tip
In practice we rely on algebraic rules to differentiate quickly rather than limits from first principles.
Basic Rules and Derivatives
| Rule | Function | Derivative |
|---|---|---|
| Linearity | af + bg | a f' + b g' |
| Constant | c | 0 |
| Power rule | x^n | n x^{n-1} (n real) |
| Exponential | e^x | e^x |
| a^x | a^x ln a | |
| Logarithm | ln x | 1/x |
| log_a x | 1/(x ln a) | |
| Trigonometric | sin x | cos x |
| cos x | -sin x | |
| tan x | sec^2 x | |
| cot x | -csc^2 x | |
| sec x | sec x tan x | |
| csc x | -csc x cot x | |
| Inverse trigonometric | arcsin x | 1/√(1-x^2) |
| arccos x | -1/√(1-x^2) | |
| arctan x | 1/(1+x^2) | |
| Product rule | fg | f'g + fg' |
| Quotient rule | f/g | (f'g - fg')/g^2 (g ≠ 0) |
| Chain rule | f(g(x)) | f'(g(x)) g'(x) |
Interactive Derivative
Type any function of $x$ below. The widget computes $f'(x)$ symbolically and provides a link to WolframAlpha for step-by-step explanations.
Worked Example
Let $f(x) = \sin^2 x \cdot \ln x$. Differentiate using product + chain rules.
- Set $u(x)=\sin^2 x$ and $v(x)=\ln x$. Then $f=uv$ and $f'=u'v+uv'$.
- $u'(x)=2\sin x\cos x=\sin(2x)$.
- $v'(x)=\tfrac{1}{x}$ (for $x>0$).
Putting it together:
$$ f'(x) = \sin(2x)\,\ln x + \frac{\sin^2 x}{x},\qquad x>0. $$You can verify this result in the widget above or on WolframAlpha.
Quick Examples
- $\dfrac{d}{dx}(x^3 - 4x) = 3x^2 - 4$.
- $\dfrac{d}{dx}(e^{2x}) = 2e^{2x}$.
- $\dfrac{d}{dx}(\ln(x^2+1)) = \dfrac{2x}{x^2+1}$ (chain rule).
- $\dfrac{d}{dx}(\sin^2 x) = 2\sin x\cos x = \sin(2x)$.
Practical Considerations
- Domains: expressions like $\ln x$ require $x>0$; denominators must be nonzero.
- Units: trigonometric derivatives assume radians.
- Structure: identify compositions to apply the chain rule systematically.
- Computation: the widget uses symbolic rules (via math.js). For didactic, step-by-step proofs, follow the WolframAlpha link.
References
- J. Stewart. Calculus: Early Transcendentals. (Cengage) — https://www.cengage.com/c/calculus-early-transcendentals-8e-stewart/
- M. Spivak. Calculus. (Publish or Perish) — https://www.mathpop.com/book.htm
- Wikipedia. Derivative — https://en.wikipedia.org/wiki/Derivative
- Wolfram MathWorld. Derivative — https://mathworld.wolfram.com/Derivative.html