Derivatives

The derivative $f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$ is the limit of the difference quotient. The difference quotient $\frac{f(x+h)-f(x)}{h}$ is the slope of a secant line through two nearby points on the graph of $f$.

As $h\to 0$, the secant line becomes the tangent line, and its slope becomes the derivative. This is the geometric heart of the derivative: zoom in far enough on any smooth curve and it looks like a straight line. The derivative measures the slope of that local line.

If this limit exists at $x=a$, we say $f$ is differentiable at $a$. Differentiability implies continuity — if $f$ is differentiable at $a$, then $f$ is also continuous at $a$. The converse is false: $|x|$ is continuous at $0$ but not differentiable (it has a sharp corner where two lines meet at different slopes).

Places where differentiability fails: sharp corners (like $|x|$), cusps (like $x^{2/3}$), vertical tangent lines (like $\sqrt[3]{x}$ at $0$), and discontinuities. At all of these points, the limit of the difference quotient either doesn't exist or is infinite.

Power rule: $\frac{d}{dx} x^n = nx^{n-1}$ for any real number $n$. This works for negative and fractional exponents too: $\frac{d}{dx} x^{-2} = -2x^{-3}$ and $\frac{d}{dx} \sqrt{x} = \frac{1}{2\sqrt{x}}$.

Constant multiple rule: $\frac{d}{dx}[cf(x)] = c f'(x)$. Constants pass through the derivative.

Sum/difference rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$. Differentiate term by term.

Constant rule: $\frac{d}{dx}[c] = 0$. The derivative of any constant is zero.

Exponentials: $\frac{d}{dx} e^x = e^x$ — the defining property of $e$. No other function equals its own derivative. More generally, $\frac{d}{dx} a^x = a^x \ln a$ (the $\ln a$ factor accounts for the base).

Logarithms: $\frac{d}{dx} \ln x = \frac{1}{x}$ for $x > 0$. This is why $\ln x$ appears everywhere in integration — it's the antiderivative of $1/x$. More generally, $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$.

Trigonometric: $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$ (note the negative sign!), $\frac{d}{dx} \tan x = \sec^2 x$.

More trig: $\frac{d}{dx} \sec x = \sec x \tan x$, $\frac{d}{dx} \csc x = -\csc x \cot x$, $\frac{d}{dx} \cot x = -\csc^2 x$. Notice the pattern: the 'co-' functions ($\cos$, $\csc$, $\cot$) always pick up a negative sign.

These formulas are non-negotiable — they must be committed to memory. Every more complex derivative ultimately reduces to combinations of these through the product, quotient, and chain rules.

When two functions are multiplied: $(fg)' = f'g + fg'$.

Think of it as: "derivative of the first times the second, plus the first times the derivative of the second."

Example: $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$.

Tip: always check if you can simplify the expression first. Sometimes you don't need the product rule at all.

For a ratio of functions: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$. The order matters — it's $f'g$ first, then subtract $fg'$.

Memory aid: "low d-high minus high d-low, over the square of what's below." Here 'high' is the numerator $f$, 'low' is the denominator $g$, and 'd-' means 'derivative of.'

Alternative: you can often rewrite $f/g$ as $f \cdot g^{-1}$ and use the product rule with the chain rule instead. For simple denominators, sometimes pure algebra avoids the quotient rule entirely.

When to definitely use the quotient rule: when the denominator is a non-trivial function (like $\cos x$, $e^x + 1$, etc.) and rewriting would be messier than just applying the formula.

The chain rule handles compositions: if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

In words: derivative of the outer function (evaluated at the inner function) times the derivative of the inner function.

Example: $\frac{d}{dx} \sin(x^3) = \cos(x^3) \cdot 3x^2$. The outer function is $\sin$, the inner function is $x^3$.

Nested chains: for $e^{\sin(x^2)}$, apply the chain rule twice: $e^{\sin(x^2)} \cdot \cos(x^2) \cdot 2x$.

The chain rule is arguably the most important rule. It appears everywhere: in implicit differentiation, related rates, and integration by substitution.

$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1-x^2}}$, valid for $|x| < 1$. The domain restriction comes from the original function: $\arcsin$ only takes inputs between $-1$ and $1$.

$\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1-x^2}}$. Notice this is exactly the negative of $\arcsin$'s derivative. This makes sense because $\arcsin x + \arccos x = \pi/2$ (they always add to a right angle), so their derivatives must be negatives of each other.

$\frac{d}{dx} \arctan x = \frac{1}{1+x^2}$. This is the most commonly used inverse trig derivative. It appears constantly in integration (the antiderivative of $\frac{1}{1+x^2}$ is $\arctan x + C$).

With the chain rule: $\frac{d}{dx} \arctan(g(x)) = \frac{g'(x)}{1+[g(x)]^2}$. For example, $\frac{d}{dx} \arctan(3x) = \frac{3}{1+9x^2}$.

Where these come from: use implicit differentiation. If $y = \arcsin x$, then $\sin y = x$. Differentiate: $\cos y \cdot y' = 1$, so $y' = 1/\cos y = 1/\sqrt{1-\sin^2 y} = 1/\sqrt{1-x^2}$.

When $y$ is defined implicitly by an equation like $x^2 + y^2 = 25$, differentiate both sides with respect to $x$.

Every time you differentiate a $y$ term, attach a $\frac{dy}{dx}$ factor (this is the chain rule in action: $y$ is a function of $x$).

Example: differentiate $x^2 + y^2 = 25$ to get $2x + 2y\frac{dy}{dx} = 0$, then solve $\frac{dy}{dx} = -\frac{x}{y}$.

Implicit differentiation is essential for curves that can't be written as $y = f(x)$, such as circles, ellipses, and other relations.

For complicated products, quotients, or expressions where the variable appears in both the base and exponent (like $x^x$), take the natural log of both sides first, then differentiate implicitly.

Why it works: $\ln$ converts products into sums ($\ln(fg) = \ln f + \ln g$), quotients into differences ($\ln(f/g) = \ln f - \ln g$), and powers into products ($\ln(f^g) = g \ln f$). These are all easier to differentiate.

The procedure: let $y = f(x)$, take $\ln$ of both sides, simplify using log properties, differentiate implicitly ($\frac{1}{y} \frac{dy}{dx} = \ldots$), then solve for $\frac{dy}{dx}$ by multiplying both sides by $y$.

This is the only way to differentiate variable-base, variable-exponent expressions like $x^{\sin x}$ or $(\ln x)^x$. Neither the power rule nor the exponential rule applies when both the base and exponent depend on $x$.

The second derivative $f''(x)$ is the derivative of $f'(x)$. It measures how the rate of change itself is changing — the 'rate of the rate.'

Notation: $f''(x)$, $\frac{d^2y}{dx^2}$, or $y''$. For higher orders: $f'''(x)$, $f^{(4)}(x)$, etc. The notation $\frac{d^2y}{dx^2}$ reminds you that you're applying $\frac{d}{dx}$ twice.

Physical meaning: if $s(t)$ is position, then $s'(t)$ is velocity (how fast position changes) and $s''(t)$ is acceleration (how fast velocity changes). A car accelerating has positive $s''$; a car braking has negative $s''$.

The second derivative tells you about concavity: $f''(x) > 0$ means concave up (the curve bends upward, like a bowl), $f''(x) < 0$ means concave down (bends downward, like a hill). This is central to the second derivative test for classifying extrema.

Higher-order derivatives appear in Taylor series: the $n$th coefficient uses $f^{(n)}(a)$, letting you approximate any smooth function with a polynomial. Some functions (like $e^x$ and $\sin x$) have patterns in their higher derivatives that make the Taylor series elegant.

The tangent line to $f$ at $x = a$ is: $y = f(a) + f'(a)(x - a)$. This is the best linear approximation near $a$.

Linearization: $f(x) \approx f(a) + f'(a)(x-a)$ for $x$ close to $a$. This is incredibly useful for quick estimates.

Example: approximate $\sqrt{4.1}$. Use $f(x) = \sqrt{x}$ at $a = 4$: $f'(x) = \frac{1}{2\sqrt{x}}$, $f'(4) = \frac{1}{4}$.

Linearization gives $\sqrt{4.1} \approx 2 + \frac{1}{4}(0.1) = 2.025$. The actual value is $2.02485...$, very close!

Differentials: $dy = f'(x)\,dx$ gives the approximate change in $y$ for a small change $dx$ in $x$.

$f'(x) > 0$ on an interval means $f$ is increasing (going uphill) there. $f'(x) < 0$ means $f$ is decreasing (going downhill). The magnitude $|f'(x)|$ tells you how steep the slope is.

Where $f'(x) = 0$, the tangent line is horizontal. These are critical points — candidates for local maxima, local minima, or inflection points.

The sign of $f'$ changing from $+$ to $-$ at a critical point signals a local maximum (function goes up then down). From $-$ to $+$ signals a local minimum (down then up). If $f'$ doesn't change sign, the critical point is neither (e.g., $f(x)=x^3$ at $x=0$ — the function flattens momentarily but keeps going in the same direction).

Reading $f'$'s graph to understand $f$: where $f'$ is above the $x$-axis, $f$ is climbing. Where $f'$ is below, $f$ is falling. The $x$-intercepts of $f'$ correspond to the peaks and valleys (or flat inflection points) of $f$. The higher $|f'|$ is, the steeper $f$ is at that point.

You can also go the other way: given the graph of $f$, sketch $f'$ by estimating the slope at various points. Steep uphill = large positive $f'$; flat = $f'$ near zero; steep downhill = large negative $f'$.

The hyperbolic functions are built from exponentials: $\sinh x = \frac{e^x - e^{-x}}{2}$ and $\cosh x = \frac{e^x + e^{-x}}{2}$.

They satisfy $\cosh^2 x - \sinh^2 x = 1$ (compare with $\cos^2 x + \sin^2 x = 1$ for trig).

Derivatives mirror trig but without the sign changes: $\frac{d}{dx} \sinh x = \cosh x$ and $\frac{d}{dx} \cosh x = \sinh x$.

Also: $\frac{d}{dx} \tanh x = \text{sech}^2 x$ where $\tanh x = \sinh x / \cosh x$.

Inverse hyperbolics have logarithmic forms: $\sinh^{-1} x = \ln(x + \sqrt{x^2+1})$, and their derivatives produce the expressions $\frac{1}{\sqrt{x^2+1}}$ and $\frac{1}{\sqrt{x^2-1}}$ that appear in integration.

Applications: catenary curves (hanging chains/cables obey $y = a\cosh(x/a)$), special relativity (rapidity), and certain differential equations.