The common derivatives cover polynomials, trigonometric functions, exponentials, and logarithms. Beyond these lie function families whose derivatives are less immediately obvious and harder to rederive on the spot. Inverse trigonometric functions produce algebraic expressions involving square roots. Hyperbolic functions mirror their trigonometric counterparts with subtle sign differences. Piecewise functions demand separate treatment on each interval with careful attention at boundaries.
Most of these derivatives are established through implicit differentiation or the inverse function formula rather than direct computation from the limit definition. Understanding the derivation builds insight; memorizing the result builds speed. Both matter.
Key Terms
Derivative— the formulas here extend the common derivatives to inverse trig, hyperbolic, and piecewise functions
Again the derivatives are negatives of each other, following from arctanx+arccotx=2π.
To derive the arctangent formula: let y=arctanx, so tany=x. Differentiating: sec2y⋅dxdy=1. Using the identity sec2y=1+tan2y=1+x2 gives dxdy=1+x21.
Unlike arcsine and arccosine, arctangent is defined for all real x—no domain restriction. The derivative 1+x21 is always positive, confirming that arctanx is strictly increasing. As x→±∞, the derivative approaches zero, reflecting the horizontal asymptotes y=±2π. The function rises everywhere but flattens out at the extremes.
This derivative appears prominently in integration: ∫1+x21dx=arctanx+C.
Inverse Trigonometric Functions — Arcsecant and Arccosecant
The derivation for arcsecant: let y=arcsecx, so secy=x. Differentiating: secytany⋅dxdy=1. Since secy=x and tany=±sec2y−1=±x2−1, the sign depends on the quadrant. The absolute value ∣x∣ in the formula resolves this: secytany=∣x∣x2−1 across the full domain.
These derivatives are less frequently encountered than the arcsine and arctangent versions. They appear primarily in integrals of the form ∫xx2−11dx and in problems involving trigonometric substitution.
The domain ∣x∣>1 reflects the range of secant: secy never takes values between −1 and 1.
Patterns in Inverse Trigonometric Derivatives
The six inverse trigonometric derivatives organize into three pairs, each pair summing to zero:
(arcsinx)′+(arccosx)′=0, both involving 1−x21
(arctanx)′+(arccotx)′=0, both involving 1+x21
(arcsecx)′+(arccscx)′=0, both involving ∣x∣x2−11
Each pair shares the same magnitude because the two functions in each pair sum to a constant (2π). The cofunction always carries the negative sign—the same pattern seen in the common trigonometric derivatives where cos, cot, and csc derivatives are negative.
Two distinct algebraic forms appear: 1−x2 for the sine/cosine pair, 1+x2 for the tangent/cotangent pair, and ∣x∣x2−1 for the secant/cosecant pair. These three expressions reappear in integration, where recognizing them triggers the corresponding inverse trigonometric antiderivative. The table below collects the three algebraic forms alongside their cofunction pairs and the identity that explains why each pair shares the same magnitude.
Algebraic form (magnitude)
Cofunction pair
Why they share the same magnitude
1 ⁄ √(1 − x2)
arcsin x ↔ arccos x
arcsin x + arccos x = π ⁄ 2, so the two derivatives sum to zero
1 ⁄ (1 + x2)
arctan x ↔ arccot x
arctan x + arccot x = π ⁄ 2, so the two derivatives sum to zero
1 ⁄ (|x| √(x2 − 1))
arcsec x ↔ arccsc x
arcsec x + arccsc x = π ⁄ 2, so the two derivatives sum to zero
Hyperbolic Functions
The hyperbolic functions are defined through exponentials:
sinhx=2ex−e−xcoshx=2ex+e−xtanhx=coshxsinhx
Their derivatives parallel the trigonometric pattern:
The critical difference from trigonometric derivatives: (coshx)′=sinhx carries no negative sign, whereas (cosx)′=−sinx does. This single sign difference propagates through all six formulas.
Each derivative is verified directly by differentiating the exponential definitions. For sinhx: dxd[2ex−e−x]=2ex+e−x=coshx. No special limits or implicit differentiation are needed—the exponential derivative does all the work.
Inverse Hyperbolic Functions
The inverse hyperbolic functions have derivatives that resemble the inverse trigonometric ones with sign changes under the radical:
Compare with inverse trigonometric derivatives: 1−x2 becomes x2+1 or x2−1, and 1+x2 becomes 1−x2. The signs inside the expressions flip.
These derivatives can be established through implicit differentiation or through the logarithmic representations. For instance, arcsinhx=ln(x+x2+1). Differentiating this logarithmic form directly using the chain rule confirms x2+11.
Inverse hyperbolic derivatives appear in integration: ∫x2+11dx=arcsinhx+C and ∫x2−11dx=arccoshx+C provide alternatives to trigonometric substitution for certain radical integrals.
Piecewise Functions
A piecewise function uses different formulas on different intervals. On each interval's interior, standard differentiation rules apply to the governing formula. The derivative of the piecewise function is itself piecewise—computed piece by piece.
The challenge lies at the boundaries. At a point x=c where the formula changes, the derivative exists only if three conditions hold: the function is continuous at c (the pieces connect), the left-hand derivative exists, and the right-hand derivative exists and equals the left-hand derivative.
Consider f(x)=x2 for x≤1 and f(x)=2x−1 for x>1. Continuity at x=1: f(1)=1 from the left, 2(1)−1=1 from the right—they match. Left derivative at 1: dxd[x2]x=1=2. Right derivative at 1: dxd[2x−1]x=1=2. Both derivatives match, so f′(1)=2.
If either continuity fails or the one-sided derivatives disagree, the function is not differentiable at that boundary. The absolute value function ∣x∣ is the canonical example: continuous at x=0, but left derivative −1 and right derivative +1 do not match. The table below names each of the three boundary conditions, what it requires, and the failure mode that appears when it is violated.
Condition at boundary x = c
What it requires
Failure mode if violated
Continuity at c
the two pieces meet: the left and right values both equal f(c)
jump discontinuity — no tangent line possible
One-sided derivatives exist
both the left-hand and right-hand derivatives at c are finite
vertical tangent or cusp — slope is infinite or undefined on one side
One-sided derivatives agree
left derivative at c equals right derivative at c
corner — e.g., |x| at 0, where the slopes are −1 and +1
Summary of Special Derivatives
The complete set of special derivative formulas:
dxd[arcsinx]=1−x21dxd[arccosx]=−1−x21
dxd[arctanx]=1+x21dxd[arccotx]=−1+x21
dxd[arcsecx]=∣x∣x2−11dxd[arccscx]=−∣x∣x2−11
dxd[sinhx]=coshxdxd[coshx]=sinhx
dxd[tanhx]=sech2xdxd[cothx]=−csch2x
dxd[sechx]=−sechxtanhxdxd[cschx]=−cschxcothx
dxd[arcsinhx]=x2+11dxd[arccoshx]=x2−11
dxd[arctanhx]=1−x21
Together with the common derivatives, these formulas cover every standard function encountered in calculus. Recognizing the algebraic forms—1−x2, 1+x2, x2±1, 1−x2—is equally important for integration, where these expressions signal inverse trigonometric or inverse hyperbolic antiderivatives. The table below assembles all 15 formulas as a single reference card, organized by category and including the domain on which each derivative is valid.