Mistakes

For large x the fastest-growing piece of each part decides; keep it, and in a quotient divide the top and the bottom by it.

Close to 0 the smaller exponent dominates; the larger one takes over only far away.

After isolating a power of x, take the matching root, and substitute the answer back to check.

Inside the function the sign is reversed: x minus 3 moves the graph right, x plus 3 moves it left.

x², x⁴ and (x − a)² are never negative; only odd exponents flip across zero.

A minus sign upstairs means a reciprocal: (1/4)^−1/2 = 4^1/2 = 2.

A power of a power multiplies the exponents, a product adds them, a quotient subtracts them, and an nth root is the power 1/n.

Set the exponential equal to the target ratio and take logarithms of both sides to bring the unknown down.

Only something that multiplies the whole top and the whole bottom can be struck out; a piece that is added cannot.

0/0 is not an answer; it says the expression has not been simplified yet.

The square root of x² is |x|, which is minus x when x is negative.

Bound the wave between −1 and 1; then the product lies between −|g| and |g|, where g is the factor in front, and both bounds tend to 0 when g does.

Cancel first: a vertical asymptote needs the denominator to vanish afterwards, a zero of the numerator is an intercept only where f itself is defined, and only the leading terms matter at infinity.

When the values grow without bound with one sign, the limit is not a number, but write +∞ or −∞: that says how it fails, and it is the answer asked for.

For a one-sided limit, check the sign each factor takes on that side.

A limit is about nearby points; the value at the point itself may differ or not exist.

Continuity at a point needs the limit and the value to exist and to agree.

A derivative at a point forces the graph to be unbroken there, but not the reverse: |x| is unbroken at 0 and has a corner, so no slope.

Zero under a root is allowed and zero in a denominator is not; check every candidate against the domain of the situation.

The degree only bounds the count; the turning points and the sign of the function there decide it.

Factor instead, or handle the case where it vanishes on its own, so that no candidate is lost.

f(a + h) minus f(a), over h, as h tends to 0: the same h in every place, and the subtraction in that order.

Carry the sign through every line and check it against the sign the answer must have.

Reread what the question asks, and check that the number you give is the thing it names, not one step before or after it.

Substitute into every part and carry each piece through to the end, constants included.

The y-intercept is the line’s height at x = 0; the height it shares with the curve is at x = a, where the two touch.

Low d-high minus high d-low, over low squared: the order in the numerator sets the sign, and the denominator is squared.

A product or quotient is not differentiated, or antidifferentiated, factor by factor: use the rule, or spot the chain rule in reverse.

xⁿ brings the n down, but a^x has the variable upstairs and differentiates to a^x log a.

The derivative of cos is minus sin; check the sign before anything else.

A number such as e³, π² or log 5 contains no x, so its derivative is 0.

After differentiating the outside, multiply by the derivative of the inside, every time.

The outside function’s derivative is taken at the output of the inside, not at x.

When both parts of a power vary with x, take logarithms first, so that each part contributes its own term.

Work from the inside out: say which function is applied first, then substitute back to check.

A multiple in the exponent becomes a power: e^{2 log 3} = (e^{log 3})² = 9.

The derivative of log f is f′/f, so multiply by f at the end.

When both pieces change, the derivative has two terms: u′v + uv′.

y depends on x, so the chain rule brings out a factor of dy/dx from everything in y that is differentiated.

Check sin and cos at the standard angles on the unit circle before substituting.

Check the table: log x differentiates to 1/x, a^x to a^x log a, and 1/x antidifferentiates to log|x|.

arcsin and arctan return angles between −π/2 and π/2 and arccos returns angles between 0 and π, so arcsin(sin θ) equals θ only when θ already lies in that range.

Put the numbers of the instant in last, after the derivative is taken, since a fixed number has no rate of change.

A quantity that shrinks has a negative derivative, so write the minus in before substituting.

Rates follow the derivative of the relation, not the relation itself: differentiate first, then substitute.

Draw the triangle and label which distance is the hypotenuse and which is a leg before writing the relation.

Before differentiating, decide what moves with time and write it in terms of the variable you track; only the truly fixed dimensions stay as numbers.

Check for 0/0 or ∞/∞ before differentiating; a top and bottom heading anywhere else, such as 1 over 0 or 0 over ∞, decide the limit directly, with a sign check when the bottom tends to 0.

sin(u)/u tends to 1 as u → 0, so the argument of the sine and the denominator must be the same; rescale so that they are.

0·∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰ can come out as anything: rewrite as a quotient and let l’Hôpital or a logarithm decide.

When f′/g′ has no limit, l’Hôpital’s rule says nothing about f/g, so settle it another way, such as dividing through by the fastest-growing term.

The sign of f′, not the sign of f, says whether the graph rises or falls.

f′ says where the graph rises and falls, f″ how it bends and where it inflects, so match the order to what is asked.

Look for where f′ is undefined as well as where it is zero, since a cusp or a corner can be a local extremum.

Test the definition instead: substitute −x and compare the result with f(x) and −f(x), and accept a period only when every term repeats after it.

A zero derivative marks a candidate: a sign change of f′ says which kind, and so does f″ when it is not 0; with no sign change it is neither.

Read whether the question wants the x at which the extreme occurs or the value there, and give that one.

Draw the picture and read the condition off it: count the sides fenced, the faces priced, whether there is a top.

The derivative is taken at the base point, the point of tangency on the curve; find that point before substituting.

The change in output is the slope times the change in input from the base point: Δy ≈ f′(a) Δx.

sin(x²) and (sin x)² are different functions; read the brackets before differentiating.

The coefficient of (x − a)^k is f^(k)(a)/k!, and the remainder after degree n divides by (n + 1)!.

Write the equation out in full before clearing fractions, and substitute the answer back to check.

The next estimate is x₀ − f(x₀)/f′(x₀): subtract the ratio, and let the signs of f(x₀) and f′(x₀) decide which way the estimate moves.

When the slope depends on the amount or the time, it has to be found again at each step, or the exact solution used; one slope held throughout is a straight line.

The law is about T minus the room temperature: the model is written for that difference, and the room is where the object ends up.

Put the initial time into the general solution and solve the equation it gives for the constant, which equals the initial value only when the solution at t = 0 reduces to the constant itself.

Draw the arrows from the sign of f between the steady states: a steady state is stable only when both arrows point at it, and a solution follows the arrow where it starts.

In a partial derivative the other letter is a constant, so ∂/∂x of x²y is 2xy and a term in y alone gives 0.

Work out D = f_xx f_yy − (f_xy)², with the mixed partial squared, and read its sign first: D < 0 is a saddle whatever f_xx says, and f_xx decides only when D > 0.

Write + C every time, and fix each constant with the condition that belongs to that function.