Difficulty: Foundational to Intermediate | Prerequisites: Limit definition of the derivative, algebra of exponents and logarithms

Big Picture

Before you can tackle the chain rule, product rule, or any integration technique, you need to know the derivatives of the elementary functions cold. This set of notes collects every derivative formula from Calc I and the new ones introduced in Calc II (exponentials, logarithms, inverse trig, hyperbolics) into one reference. Commit these to memory: they are the building blocks for every problem that follows.

TL;DR

Every standard function (polynomials, trig, exponentials, logs, inverse trig, hyperbolics) has a derivative formula you should know by heart. The power rule handles all polynomial terms. Trig derivatives come in pairs with sign patterns worth memorising. Exponential and log derivatives introduce ln(a) as a conversion factor.

Key Terms

Power rule

The formula d/dx[xⁿ] = n·xⁿ⁻¹, valid for any real exponent n. Think of it as: bring the exponent down as a coefficient, then reduce the exponent by one.

Exponential function (base a)

A function of the form aˣ. Its derivative is ln(a)·aˣ. In simple terms, differentiating an exponential gives back the same exponential, scaled by the natural log of the base.

Natural exponential (base e)

The special case eˣ, whose derivative is itself. This is what makes e the "natural" base for calculus.

Logarithmic function

The inverse of the exponential. d/dx[logₐ(x)] = 1/(ln(a)·x). For natural log: d/dx[ln(x)] = 1/x.

Inverse trigonometric functions (arcsin, arccos, arctan)

The "undo" functions for sin, cos, tan. Their derivatives involve square roots and squared terms, and they come with domain restrictions.

Hyperbolic functions (sinh, cosh, tanh, etc.)

Functions built from exponentials: sinh(x) = (eˣ - e⁻ˣ)/2, cosh(x) = (eˣ + e⁻ˣ)/2. Their derivative rules mirror the standard trig rules with minor sign differences.

Power Rule and Constants

\frac{d}{dx}[x^n] = n \cdot x^{n-1}, \quad \text{where } n \text{ is any real number}

This works for integer exponents, fractional exponents (roots), and negative exponents (reciprocals). Examples:

• d/dx[x⁵] = 5x⁴

• d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2) = 1/(2√x)

• d/dx[1/x³] = d/dx[x⁻³] = -3x⁻⁴

Derivative of a constant:

\frac{d}{dx}[c] = 0

A constant has no rate of change. This is the simplest derivative and one students sometimes forget to apply when a constant sits inside a more complex expression.

Trigonometric Function Derivatives

The six standard trig derivatives come in three pairs. The "co-" function in each pair picks up a negative sign.

\frac{d}{dx}[\sin(x)] = \cos(x) \qquad \frac{d}{dx}[\cos(x)] = -\sin(x)

\frac{d}{dx}[\tan(x)] = \sec^2(x) \qquad \frac{d}{dx}[\cot(x)] = -\csc^2(x)

\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x) \qquad \frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)

The pattern: derivatives of sin, tan, and sec are positive. Derivatives of cos, cot, and csc carry a minus sign. The tan/cot pair produces squared functions. The sec/csc pair produces products of two trig functions.

Exponential and Logarithmic Derivatives

Exponential (general base a):

\frac{d}{dx}[a^x] = \ln(a) \cdot a^x

Exponential (base e):

\frac{d}{dx}[e^x] = e^x

The natural exponential is its own derivative. This property is the reason e appears so often in differential equations and applied mathematics.

Logarithmic (general base a):

\frac{d}{dx}[\log_a(x)] = \frac{1}{\ln(a)} \cdot \frac{1}{x}

Natural logarithm (base e):

\frac{d}{dx}[\ln(x)] = \frac{1}{x}

Notice the symmetry: ln(a) appears as a multiplier in the exponential derivative and as a divisor in the log derivative. When the base is e, ln(e) = 1 and the factor disappears.

Inverse Trigonometric Derivatives

These appear frequently in Calc II integration problems (they show up as antiderivatives).

\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}

\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}, \quad -1 < x < 1

\frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 - x^2}}, \quad -1 < x < 1

Note that arcsin and arccos have the same denominator but opposite signs. The domain restriction (-1, 1) comes from the original range of sin and cos. arctan has no such restriction and is defined for all real x.

Hyperbolic Function Derivatives

Hyperbolic functions are defined in terms of exponentials:

\sinh(x) = \frac{e^x - e^{-x}}{2} \qquad \cosh(x) = \frac{e^x + e^{-x}}{2}

The remaining hyperbolic functions (tanh, coth, sech, csch) are built from sinh and cosh the same way standard trig functions are built from sin and cos.

Their derivatives mirror the trig pattern closely, with one important sign difference:

• d/dx[sinh(x)] = cosh(x)

• d/dx[cosh(x)] = sinh(x) (positive, unlike d/dx[cos(x)] = -sin(x))

• d/dx[tanh(x)] = sech²(x)

• d/dx[coth(x)] = -csch²(x)

• d/dx[sech(x)] = -sech(x)·tanh(x)

• d/dx[csch(x)] = -csch(x)·coth(x)

The key difference from regular trig: d/dx[cosh(x)] = +sinh(x), not -sinh(x). The "co-function gets a negative" rule breaks here. This is the most common mix-up in exam problems involving hyperbolics.

Common Misconceptions

• Students often think d/dx[eˣ] = x·eˣ⁻¹, applying the power rule to the exponent instead of the base. The power rule is for xⁿ (variable base, constant exponent). Exponential functions have a constant base and variable exponent, which is a fundamentally different situation.

• Forgetting the ln(a) factor when differentiating aˣ for bases other than e. d/dx[2ˣ] = ln(2)·2ˣ, not just 2ˣ.

• Mixing up the sign on d/dx[cosh(x)]. Unlike d/dx[cos(x)] = -sin(x), the hyperbolic version has no minus sign: d/dx[cosh(x)] = +sinh(x).

• Confusing arcsin with 1/sin (i.e., csc). arcsin is the inverse function, csc is the reciprocal. They have completely different derivatives.

Why It Matters / Exam Flags

⚠️ The trig derivative pairs and their signs are tested constantly. Make a flashcard set.

⚠️ Inverse trig derivatives reappear in Calc II as integration results. Recognising that 1/(1 + x²) is the derivative of arctan(x) is essential for integration by substitution.

⚠️ Hyperbolic functions may appear in problems on catenary curves, differential equations, or integration. Know the definitions and derivative formulas.

⚠️ Be ready to differentiate compound expressions like ln(x²) or e^(3x) using these formulas combined with the chain rule.

Quick Self-Test

1. d/dx[x⁻⁷] = ?

1. d/dx[cos(x)] = ?

1. d/dx[eˣ] = ?

1. d/dx[ln(x)] = ?

1. d/dx[arctan(x)] = ?

Answers: 1. -7x⁻⁸. 2. -sin(x). 3. eˣ. 4. 1/x. 5. 1/(1 + x²).

Practice Q&A

Q: Differentiate f(x) = 5eˣ + 3ln(x).

A: f'(x) = 5eˣ + 3/x.

Q: Find the derivative of y = log₁₀(x).

A: dy/dx = 1/(ln(10)·x). The ln(10) factor converts from base 10 to the natural base.

Q: What is d/dx[sec(x)] and how does it relate to d/dx[csc(x)]?

A: d/dx[sec(x)] = sec(x)·tan(x). d/dx[csc(x)] = -csc(x)·cot(x). The csc derivative follows the same structure but with a negative sign and the "co-" versions of each function.

Q: Differentiate y = arcsin(x). On what interval is this derivative valid?

A: dy/dx = 1/√(1 - x²), valid for -1 < x < 1. At x = ±1 the denominator is zero, so the derivative is undefined.

Q: True or false: d/dx[cosh(x)] = -sinh(x).

A: False. d/dx[cosh(x)] = +sinh(x). The sign difference from standard trig is a common trap.

Connections to Other Topics

These formulas form the foundation for the differentiation rules (product, quotient, chain) covered in the next set of notes. In Calc II, the inverse trig derivatives reappear as standard integration results, and the exponential/log derivatives are central to integration by parts and solving separable differential equations.

Related Terms / Search Tags

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