Calculus reference

What kind of answer is requested?

Expression	What it returns	For f(x) = x²
$f(3)$	An output value	9
$f'(x)$	A function describing the rate of change	$2x$
$f'(3)$	The rate at one input	6
$f'(3)h$	A predicted output change for a small input change h	$6h$
$f(3)+f'(3)h$	A predicted new output	$9+6h$
$\int_0^1 f(x)\,dx$	Accumulated signed area over an interval	$1/3$
$\int f(x)\,dx$	A family of antiderivatives on an interval	$x^3/3+C$

Derivative rules

Assume u and v are differentiable where the rule is applied, and c is constant.

Operation	Rule	Example
Sum	$(u+v)'=u'+v'$	$(x^2+x)'=2x+1$
Constant multiple	$(cu)'=cu'$	$(3x^2)'=6x$
Power	$(x^n)'=nx^{n-1}$	$(x^3)'=3x^2$
Product	$(uv)'=u'v+uv'$	$(x^2e^x)'=2xe^x+x^2e^x$
Composition	$(g(u(x)))'=g'(u(x))u'(x)$	$((3x+1)^2)'=2(3x+1)\cdot3$
Quotient	$(u/v)'=(u'v-uv')/v^2$	$(1/x)'=-1/x^2$

The quotient requires $v\ne0$. The power rule applies to real n on $x>0$; particular powers have larger domains. The derivative of $\sqrt{x}$ is $1/(2\sqrt{x})$ for $x>0$, not at zero.

Other useful derivatives are $(e^x)'=e^x$, $(\ln x)'=1/x$ for $x>0$, and $(a^x)'=(\ln a)a^x$ for $a>0$. The formulas $(\sin x)'=\cos x$ and $(\cos x)'=-\sin x$ use radians.

A zero derivative is a candidate

An interior local maximum or minimum of a differentiable function has $f'(x)=0$. The converse does not hold: $x^3$ has a zero derivative at zero and keeps increasing.

At a stationary point, a positive second derivative gives a local minimum, a negative one gives a local maximum, and zero gives no verdict. Also check boundaries and points where the derivative does not exist. For a global claim, compare the whole stated domain.

Integrals

For continuous f on $[a,b]$ and an antiderivative F with $F'=f$,

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

Integrand	An antiderivative	Condition
$x^n$	$x^{n+1}/(n+1)$	$n\ne-1$, on an interval where the power rule applies
$1/x$	$\ln\lvert x\rvert$	Interval does not cross zero
$e^x$	$e^x$	All real x
c	cx	c constant

A definite integral counts area below the axis negatively. The total geometric area is $\int_a^b |f(x)|\,dx$ when $a<b$.

The unweighted average height is $\frac1{b-a}\int_a^b f(x)\,dx$. An expectation instead weights by a probability distribution; for a continuous X, $\mathbb E[g(X)]=\int g(x)f_X(x)\,dx$ when the expectation exists.

Partial derivatives and gradients

For $f(x,y)=x^2y+y$, hold the other input fixed:

$$\frac{\partial f}{\partial x}=2xy,\qquad \frac{\partial f}{\partial y}=x^2+1.$$

At $(2,3)$, the gradient is $(12,5)$. A small step $(h,k)$ gives the linear prediction $\Delta f\approx12h+5k$.

For steps of the same Euclidean length, the nonzero gradient gives the direction of greatest first-order increase. A sufficiently small step against it decreases a differentiable function:

$$\theta_{\mathrm{new}}=\theta-\eta\nabla L(\theta).$$

Here θ is the vector of parameters, L the loss, and η a positive learning rate. A large step can increase the loss. A zero gradient alone does not prove a minimum.

Three notation questions

Why can h approach zero after we divided by h?

The quotient is evaluated for nonzero h. Its limit describes the value those quotients approach. For $x^2$ at 3, the quotient is $6+h$ and its limit is 6. We never evaluate $0/0$.

What does dx mean?

In the integral notation used here, dx identifies the integration variable. The finite rectangle widths are Δx; the integral is their limiting sum. In df/dx, the notation identifies a derivative. Differential notation has further uses, but treating dx as literally zero does not justify these operations.

What is the difference between d and ∂?

A partial derivative holds other independent inputs fixed. A total derivative follows how the input actually changes. For $f(x,y)=xy$, the partial derivative in x is y. If instead y also changes as $y=x$, then $f(x,x)=x^2$ and the derivative along that path is $2x$. State what is fixed before differentiating.

Practice these distinctions or apply a gradient to a fitted line.