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.

As $h\to 0$, the secant line becomes the tangent line, and its slope becomes the derivative.

If this limit exists at $x=a$, we say $f$ is differentiable at $a$. 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).

Places where differentiability fails: sharp corners (like $|x|$), vertical tangent lines (like $\sqrt[3]{x}$ at $0$), and discontinuities.

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 only function equal to its own derivative). More generally, $\frac{d}{dx} a^x = a^x \ln a$.

Logarithms: $\frac{d}{dx} \ln x = \frac{1}{x}$. More generally, $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$.

Trigonometric: $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$, $\frac{d}{dx} \tan x = \sec^2 x$.

More trig: $\frac{d}{dx} \sec x = \sec x \tan x$, $\frac{d}{dx} \csc x = -\csc x \cot x$, $\frac{d}{dx} \cot x = -\csc^2 x$.

Memorize these. They are the building blocks for every derivative you will ever compute.

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: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

Memory aid: "low d-high minus high d-low, over the square of what's below."

Example: $\frac{d}{dx}\frac{x^2}{\cos x} = \frac{2x\cos x - x^2(-\sin x)}{\cos^2 x} = \frac{2x\cos x + x^2\sin x}{\cos^2 x}$.

Alternative: you can often rewrite $f/g$ as $f \cdot g^{-1}$ and use the product rule instead. Use whichever feels easier.

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 $|x| < 1$.

$\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1-x^2}}$.

$\frac{d}{dx} \arctan x = \frac{1}{1+x^2}$. This one appears frequently in integration.

With the chain rule: $\frac{d}{dx} \arctan(3x) = \frac{3}{1+(3x)^2} = \frac{3}{1+9x^2}$.

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 like $x^x$, take the natural log of both sides first.

Example: find $\frac{d}{dx} x^x$. Let $y = x^x$, then $\ln y = x \ln x$.

Differentiate implicitly: $\frac{1}{y}\frac{dy}{dx} = \ln x + 1$.

Solve: $\frac{dy}{dx} = x^x(\ln x + 1)$.

This technique turns products into sums and powers into products, making differentiation much simpler.

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

Notation: $f''(x)$, $\frac{d^2y}{dx^2}$, or $y''$. For higher orders: $f'''(x)$, $f^{(4)}(x)$, etc.

Physical meaning: if $s(t)$ is position, then $s'(t)$ is velocity and $s''(t)$ is acceleration.

The second derivative tells you about concavity: $f''(x) > 0$ means concave up (like a bowl), $f''(x) < 0$ means concave down (like an arch).

Higher-order derivatives appear in Taylor series: you need $f'(a), f''(a), f'''(a), \ldots$ to build the polynomial approximation.

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 there. $f'(x) < 0$ means $f$ is decreasing.

Where $f'(x) = 0$, the tangent line is horizontal. These are critical points: potential maxima or minima.

The sign of $f'(x)$ changing from $+$ to $-$ at a critical point signals a local maximum. From $-$ to $+$ signals a local minimum.

If $f'(x)$ does not change sign, the critical point is neither a max nor a min (e.g., $f(x)=x^3$ at $x=0$).

The graph of $f'(x)$ tells you the steepness and direction of $f$. Where $f'$ crosses zero, $f$ has a horizontal tangent.