Derivatives
Derivative from First Principles
The derivative of a function measures its instantaneous rate of change — the slope of the tangent to the curve at a point. It is defined as a limit of average rates of change, known as the first principle (or $ab\,initio$) definition:
The quotient $\dfrac{f(x+h) - f(x)}{h}$ is the slope of the secant joining the points $(x,\, f(x))$ and $(x+h,\, f(x+h))$. As $h \to 0$ the second point slides towards the first, and the secant pivots into the tangent; its slope is the derivative.
Derivatives from first principles. Applying the definition to the basic functions gives results worth knowing on sight:
| Function $f(x)$ | Derivative $f'(x)$ |
|---|---|
| $k$ (constant) | $0$ |
| $x^n$ | $n\,x^{n-1}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $e^x$ | $e^x$ |
| $\ln x$ | $\dfrac{1}{x}$ |
Rules of differentiation. Once the basics are known, four rules let you combine them. For differentiable functions $u$ and $v$:
A frequent slip is to assume $(uv)' = u'v'$ — it is not. The product rule keeps both cross terms, and the quotient rule subtracts (order matters: it is $u'v - uv'$ on top, never the reverse).
Deeper Insight — the derivative is a limit of slopes: Everything in this topic rests on one picture. The fraction $\dfrac{f(x+h)-f(x)}{h}$ is just "rise over run" between two points on the curve — an average rate of change across a step of width $h$. Shrinking $h$ towards zero is what converts that average into an instantaneous rate, and geometrically it is the secant line rotating until it rests as the tangent. This is also why first principles repeatedly throw up the standard limits of the previous topic: differentiating $\sin x$ forces $\lim_{h\to 0}\tfrac{\sin h}{h}=1$ to surface, and differentiating $e^x$ forces $\lim_{h\to 0}\tfrac{e^h-1}{h}=1$. The table of derivatives and the four rules are simply the compiled, reusable output of this limit process; understanding that they come from a limit — rather than appearing by decree — is what makes the whole of differential calculus cohere.
- Apply the definition: $f'(x) = \lim_{h \to 0}\dfrac{(x+h)^2 - x^2}{h}$.
- Expand the numerator: $(x+h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh + h^2$.
- $f'(x) = \lim_{h \to 0}\dfrac{2xh + h^2}{h} = \lim_{h \to 0}(2x + h)$.
- Let $h \to 0$: $f'(x) = 2x$.
Answer: $f'(x) = 2x$.
- $f'(x) = \lim_{h \to 0}\dfrac{\sin(x+h) - \sin x}{h}$.
- Use $\sin(x+h) - \sin x = 2\cos\!\left(x + \dfrac{h}{2}\right)\sin\dfrac{h}{2}$.
- $f'(x) = \lim_{h \to 0}\cos\!\left(x + \dfrac{h}{2}\right)\cdot\dfrac{\sin(h/2)}{h/2}$.
- As $h \to 0$: $\dfrac{\sin(h/2)}{h/2} \to 1$ and $\cos\!\left(x+\dfrac{h}{2}\right) \to \cos x$.
Answer: $f'(x) = \cos x$.
- Differentiate term by term using $(x^n)' = n\,x^{n-1}$ and the constant-multiple rule.
- $\dfrac{d}{dx}(4x^3) = 12x^2$, $\dfrac{d}{dx}(-7x^2) = -14x$, $\dfrac{d}{dx}(5x) = 5$, $\dfrac{d}{dx}(-9) = 0$.
- Add: $\dfrac{dy}{dx} = 12x^2 - 14x + 5$.
Answer: $\dfrac{dy}{dx} = 12x^2 - 14x + 5$.
- Let $u = x^2$ and $v = \sin x$; then $u' = 2x$ and $v' = \cos x$.
- Product rule: $\dfrac{dy}{dx} = u'v + uv' = 2x\sin x + x^2\cos x$.
Answer: $\dfrac{dy}{dx} = 2x\sin x + x^2\cos x$.
- Let $u = \sin x$ and $v = x$; then $u' = \cos x$ and $v' = 1$.
- Quotient rule: $\dfrac{dy}{dx} = \dfrac{u'v - uv'}{v^2} = \dfrac{x\cos x - \sin x}{x^2}$.
Answer: $\dfrac{dy}{dx} = \dfrac{x\cos x - \sin x}{x^2}$.
- The slope of the tangent is the derivative evaluated at the point.
- $\dfrac{dy}{dx} = 3x^2$.
- At $x = 2$: $3(2)^2 = 3 \times 4 = 12$.
Answer: The slope of the tangent at $x = 2$ is $12$.
- First principle: $f'(x) = \lim_{h \to 0}\dfrac{f(x+h) - f(x)}{h}$ — the limit of the secant slope.
- Key derivatives: $(x^n)' = n\,x^{n-1}$, $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$.
- Product rule: $(uv)' = u'v + uv'$ — never $u'v'$.
- Quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$ — order on top matters.
- The derivative at a point is the slope of the tangent (instantaneous rate of change) there.
Rules of Differentiation
Power rule d/dx(xⁿ) = nxⁿ⁻¹. Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v².
- d/dx(xⁿ) = nxⁿ⁻¹.
- Use product and quotient rules for combinations.
Derivatives of Standard Functions
Know d/dx: sinx = cosx, cosx = −sinx, eˣ = eˣ, ln x = 1/x. Combine with the rules above.
- (sinx)′ = cosx, (cosx)′ = −sinx.
- (eˣ)′ = eˣ, (ln x)′ = 1/x.