Differentiability & Differentiation
The slope of a curve at a single point — turning the limit of a secant into the derivative, and a handful of rules into a complete differentiation toolkit
- The derivative as a limit of secant slopes, and as the slope of the tangent.
- The link between differentiability and continuity, via left- and right-hand derivatives.
- The table of standard derivatives for powers, trigonometric, exponential and logarithmic functions.
- The product, quotient and chain rules that combine them.
- Implicit, logarithmic, parametric and inverse-function differentiation.
- The inverse-trigonometric derivatives and higher-order derivatives.
The Derivative as a Limit
Take two points on a curve and the line through them — a secant — has a definite slope. Slide the second point toward the first and the secant rotates into the tangent. The slope it settles on is the derivative, the instantaneous rate of change of \(f\) at that point.
The quotient inside is the secant slope between \(x\) and \(x+h\); its limit, when it exists, is the slope of the tangent. The same quantity is the instantaneous rate of change of \(y\) with respect to \(x\).
Differentiability & Continuity
A function is differentiable at \(a\) only if the one-sided derivatives agree. Every differentiable function is continuous — but the converse fails, as a sharp corner shows.
The left- and right-hand derivatives are \(\lim_{h\to0^-}\tfrac{f(a+h)-f(a)}{h}\) and \(\lim_{h\to0^+}\tfrac{f(a+h)-f(a)}{h}\); the function is differentiable at \(a\) only if they are equal. Differentiability forces continuity, but continuity alone is not enough — \(|x|\) is continuous yet not differentiable at \(0\).
Derivatives of Standard Functions
Differentiating from first principles once for each family yields a table that, combined with the rules ahead, covers almost everything you will meet.
Trigonometric: \((\sin x)'=\cos x\), \((\cos x)'=-\sin x\), \((\tan x)'=\sec^2x\). Exponential base \(a\): \((a^x)'=a^x\ln a\). And \((\sqrt{x})'=\tfrac{1}{2\sqrt{x}}\) is just the power rule with \(n=\tfrac12\).
\(\dfrac{d}{dx}\big(c\,f\big)=c\,f'\) — constants pass straight through.
\(\dfrac{d}{dx}\big(f\pm g\big)=f'\pm g'\) — differentiate term by term.
\(\dfrac{d}{dx}(c)=0\) — a constant has zero rate of change.
Product & Quotient Rules
The derivative of a product is not the product of the derivatives. Two dedicated rules handle products and quotients of functions.
Read the product rule as "derivative of the first times the second, plus the first times derivative of the second." For the quotient, the order in the numerator matters — it is \(f'g-fg'\), not the reverse — and the denominator is squared.
The Chain Rule
For a composite function — a function of a function — differentiate the outer function at the inner one, then multiply by the derivative of the inner. This is the single most-used rule in all of calculus.
In Leibniz notation, if \(y=f(u)\) and \(u=g(x)\), then \(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}\) — the rates multiply. So \(\dfrac{d}{dx}\sin(3x^2+1)=\cos(3x^2+1)\cdot6x\).
Implicit & Logarithmic Differentiation
When \(y\) is tangled into an equation with \(x\) rather than solved for, differentiate implicitly, treating \(y\) as a function of \(x\) and applying the chain rule to every \(y\) term. When the function is a tower of powers or a long product, take logarithms first.
For an implicit relation, each \(y\) contributes a factor \(\tfrac{dy}{dx}\); collect those terms and solve. For logarithmic differentiation, take \(\ln\) of both sides to turn powers into products and products into sums, then differentiate — indispensable for \(y=x^x\) and similar.
Parametric & Inverse Differentiation
When a curve is given by \(x=x(t),\ y=y(t)\), the slope comes from dividing the two rates. And the derivative of an inverse function is the reciprocal of the original derivative.
The parametric formula follows from the chain rule with parameter \(t\). The inverse-function rule says the slope of \(f^{-1}\) at a point is the reciprocal of the slope of \(f\) at the corresponding point — the graphs being reflections in \(y=x\).
These follow from the inverse rule: \(\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}\), \(\dfrac{d}{dx}\cos^{-1}x=-\dfrac{1}{\sqrt{1-x^2}}\), \(\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}\), and \(\dfrac{d}{dx}\cot^{-1}x=-\dfrac{1}{1+x^2}\).
Higher-Order Derivatives
Differentiating the derivative gives the second derivative, and so on. The second derivative measures how the slope itself is changing — the curvature that the next chapter exploits.
Each further differentiation drops the polynomial degree by one, so an \(n\)-th degree polynomial has zero \((n{+}1)\)-th derivative. The notations \(f'',\ y'',\ \tfrac{d^2y}{dx^2}\) all mean the same thing.
Putting It to Work
Problem. Differentiate \(f(x)=x^2\) using the definition.
Solution. Form the difference quotient and take the limit:
Problem. Show that \(f(x)=|x|\) is continuous but not differentiable at \(0\).
Solution. It is continuous since \(\lim_{x\to0}|x|=0=f(0)\). But the one-sided derivatives differ:
Problem. Differentiate \(y=\dfrac{x^2+1}{x-1}\).
Solution. With \(f=x^2+1,\ g=x-1\), apply \(\dfrac{f'g-fg'}{g^2}\):
Problem. Differentiate \(y=\sin(3x^2+1)\).
Solution. Outer derivative \(\cos\) at the inner, times the inner's derivative \(6x\):
Problem. Differentiate \(y=x^x\).
Solution. Take \(\ln\) of both sides, then differentiate implicitly:
Problem. Find \(\dfrac{dy}{dx}\) for \(x^2+y^2=25\).
Solution. Differentiate both sides with respect to \(x\), using the chain rule on \(y^2\):
Chapter Summary
\(f'(x)=\lim_{h\to0}\tfrac{f(x+h)-f(x)}{h}\) — slope of the tangent.
Needs \(\text{LHD}=\text{RHD}\); implies continuity, not conversely.
\((x^n)'=nx^{n-1}\), \((\sin x)'=\cos x\), \((e^x)'=e^x\), \((\ln x)'=\tfrac1x\).
Product \(f'g+fg'\), quotient \(\tfrac{f'g-fg'}{g^2}\), chain \(f'(g)\,g'\).
Implicit, logarithmic, parametric \(\tfrac{dy/dt}{dx/dt}\), inverse \(\tfrac{1}{dx/dy}\).
\(f''=\tfrac{d}{dx}(f')\); inverse-trig derivatives from the inverse rule.
Problems
Name the structure first — power, product, quotient, composite, implicit — then apply the matching rule. Difficulty rises down the list.
- Differentiate \(y=x^5-3x^2+7\).
- Differentiate \(y=\sqrt{x}+\dfrac1x\).
- Differentiate \(y=x^3e^x\).
- Differentiate \(y=\dfrac{\sin x}{x}\).
- Differentiate \(y=\cos(5x^3)\).
- Differentiate \(y=\ln(x^2+1)\).
- Differentiate \(y=e^{\sqrt{x}}\).
- If \(y=x^{\sin x}\), find \(\dfrac{dy}{dx}\).
- For the curve \(x=t^2,\ y=t^3\), find \(\dfrac{dy}{dx}\).
- Find \(\dfrac{dy}{dx}\) for \(4x^2+9y^2=36\).
- Differentiate \(y=\tan^{-1}x+\sin^{-1}x\).
- Find the second derivative of \(y=x^4-3x^3+2x\).