CALCULUS · INSTANTANEOUS CHANGE

Derivatives

A derivative measures how a function changes at an instant. It is simultaneously a limit, the slope of a tangent line, a local linear approximation and a rate of change.

slope = f′(a)tangent

1. Definition from a limit

The derivative of \(f\) at \(a\) is defined by the limit of secant slopes

\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]

if this limit exists and is finite. The quotient compares the change in output with the change in input over an interval of length \(h\). As \(h\to0\), the secant slope approaches the instantaneous slope.

Differentiability implies continuity. If \(f'(a)\) exists, then \(f\) is continuous at \(a\). The converse is false: \(f(x)=|x|\) is continuous at \(0\) but not differentiable there.

2. Four meanings of a derivative

Tangent slope\(f'(a)\) is the slope of the tangent line to \(y=f(x)\) at \((a,f(a))\).
Instantaneous rateIf \(s(t)\) is position, then \(s'(t)\) is instantaneous velocity.
Local linear modelNear \(a\), \(f(x)\approx f(a)+f'(a)(x-a)\).
SensitivityThe derivative estimates how much the output changes per small unit change in input.

Tangent-line equation

\[y=f(a)+f'(a)(x-a)\]

3. Derivative from first principles

For \(f(x)=x^2\), compute the derivative at an arbitrary \(x\):

\[\begin{aligned}f'(x)&=\lim_{h\to0}\frac{(x+h)^2-x^2}{h}\\&=\lim_{h\to0}\frac{2xh+h^2}{h}\\&=\lim_{h\to0}(2x+h)=2x.\end{aligned}\]

This derivation explains why the power rule gives \(\frac{d}{dx}x^2=2x\); the rule is not merely a memorized pattern.

4. Core differentiation rules

RuleDerivative
Constant\(\frac{d}{dx}c=0\)
Power\(\frac{d}{dx}x^n=nx^{n-1}\)
Constant multiple\((cf)'=cf'\)
Sum\((f+g)'=f'+g'\)
Product\((fg)'=f'g+fg'\)
Quotient\(\left(\frac fg\right)'=\frac{f'g-fg'}{g^2}\), \(g\neq0\)

Common elementary derivatives

Exponential\((e^x)'=e^x\), \((a^x)'=a^x\ln a\)
Logarithm\((\ln x)'=1/x\), \(x>0\)
Trigonometric\((\sin x)'=\cos x\), \((\cos x)'=-\sin x\)
Tangent\((\tan x)'=\sec^2x\) where defined

5. The chain rule

When one function is nested inside another, differentiate the outer function and multiply by the derivative of the inner function:

\[(f\circ g)'(x)=f'(g(x))\,g'(x).\]

For example, with \(y=(3x^2+1)^5\), the outer function is \(u^5\) and the inner function is \(u=3x^2+1\):

\[y'=5(3x^2+1)^4\cdot6x=30x(3x^2+1)^4.\]
Common error: differentiating the outside but forgetting the factor \(g'(x)\). The derivative of \((g(x))^5\) is not just \(5(g(x))^4\).

6. Worked examples

A. Polynomial

Differentiate \(f(x)=4x^3-5x^2+7x-2\).

\[f'(x)=12x^2-10x+7.\]

B. Product rule

For \(f(x)=x^2e^x\):

\[f'(x)=2xe^x+x^2e^x=e^x(x^2+2x).\]

C. Quotient rule

For \(f(x)=\frac{x^2+1}{x}\), \(x\neq0\):

\[f'(x)=\frac{2x\cdot x-(x^2+1)}{x^2}=1-\frac1{x^2}.\]

D. Tangent line

For \(f(x)=x^2\) at \(x=2\), \(f(2)=4\) and \(f'(2)=4\). Thus

\[y-4=4(x-2)\quad\Rightarrow\quad y=4x-4.\]

7. Where derivatives fail to exist

A derivative may fail to exist when the local slope does not settle to one finite value.

CornerFor \(|x|\) at \(0\), the left slope is \(-1\) and the right slope is \(+1\).
CuspSome curves approach vertical slopes with opposite signs.
Vertical tangentThe slope may grow without bound rather than converge to a finite number.
DiscontinuityA function cannot be differentiable at a point where it is discontinuous.

8. Interactive polynomial derivative tool

For \(f(x)=ax^3+bx^2+cx+d\), calculate \(f'(x)\), the slope at \(x_0\), and the tangent line.

9. Practice

  1. Use the limit definition to find the derivative of \(f(x)=3x+2\).
  2. Differentiate \(f(x)=x^4-2x^2+5\).
  3. Differentiate \(f(x)=(2x-1)^6\).
  4. Find the tangent line to \(f(x)=x^3\) at \(x=1\).
Solutions

1) \(f'(x)=3\). 2) \(f'(x)=4x^3-4x\). 3) \(f'(x)=12(2x-1)^5\). 4) \(f(1)=1\), \(f'(1)=3\), so \(y=3x-2\).

10. What comes next?

Derivatives describe local change. The next major idea is accumulation: definite integrals add infinitely many tiny contributions, and the Fundamental Theorem of Calculus links integration back to differentiation.

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