Differentiation Formulas

A comprehensive reference to the fundamental rules of derivatives in calculus.

Differentiation is a fundamental operation in calculus that concerns the rate at which a quantity changes. If $y = f(x)$ is a function, the derivative of $y$ with respect to $x$ is denoted as $f'(x)$, $dy/dx$, or $y'$. Geometrically, the derivative represents the slope of the tangent line to the graph of the function at any given point.

Below is a curated list of essential differentiation formulas that serve as the building blocks for solving complex calculus problems.

1. Basic Differentiation Rules

These are the simplest formulas, often dealing with polynomial functions and constants. Every student of calculus must memorize these rules as they are used constantly in conjunction with other formulas.

Constant Rule

&frac;d;
dx (c) = 0

The derivative of any constant value $c$ is zero. Since a constant does not change as $x$ changes, its rate of change is zero.

Power Rule

&frac;d;
dx (xⁿ) = nx^n-1

To differentiate $x$ raised to a power $n$, bring the exponent $n$ down to the front as a coefficient, and subtract 1 from the exponent. This applies to any real number $n$.

Constant Multiple Rule

&frac;d;
dx [c · f(x)] = c · &frac;d;
dx [f(x)]

Constants can be pulled out of the differentiation process. If a function is multiplied by a constant, the derivative is simply that constant multiplied by the derivative of the function.

Sum and Difference Rules

&frac;d;
dx [f(x) ± g(x)] = f'(x) ± g'(x)

The derivative of a sum (or difference) of functions is equal to the sum (or difference) of their derivatives. This allows us to break down complex polynomials into smaller, manageable terms.

2. Product and Quotient Rules

When functions are multiplied or divided by one another, we cannot simply differentiate them individually and combine them. Specific rules must be applied.

The Product Rule

&frac;d;
dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

To find the derivative of the product of two functions, take the derivative of the first function and multiply it by the second, then add the first function multiplied by the derivative of the second.

The Quotient Rule

&frac;d;2

To differentiate a function divided by another function, we use this formula. A common mnemonic to remember it is "Low d-Hi minus Hi d-Low, over the square of what's below."

3. The Chain Rule

The Chain Rule is arguably the most important differentiation rule. It is used when dealing with composite functionsfunctions nested within other functions, such as $\sin(x^2)$.

Chain Rule

&frac;d;
dx [f(g(x))] = f'(g(x)) · g'(x)

Differentiate the outer function $f$ (leaving the inner function $g(x)$ alone), then multiply by the derivative of the inner function $g(x)$. This concept extends to functions nested three or more layers deep.

Note: In Leibniz notation, the Chain Rule is extremely intuitive:

&frac;dy;
dx = &frac;dy;

4. Trigonometric Derivatives

Trigonometric functions appear frequently in physics and engineering. Their derivatives follow a cyclical pattern.

Function	Derivative
&sin;(x)	&cos;(x)
&cos;(x)	-&sin;(x)
&tan;(x)	&sec;²(x)
&cot;(x)	-&csc;²(x)
&sec;(x)	&sec;(x)&tan;(x)
&csc;(x)	-&csc;(x)&cot;(x)

5. Exponential and Logarithmic Derivatives

Exponential functions (base $e$) and logarithmic functions have unique properties in calculus. The natural logarithm $\ln(x)$ and the natural exponential function $e^x$ are inverse functions.

Natural Exponential

&frac;d;x) = e^x

The function $e^x$ is unique in calculus because its derivative is equal to itself. This makes it incredibly useful for modeling growth and decay.

General Exponential

&frac;d;x) = a^x &ln;(a)

For any base $a$ (where $a > 0$ and $a \neq 1$), the derivative involves multiplying the original function by the natural logarithm of the base.

Natural Logarithm

&frac;d;x

The derivative of the natural logarithm of $x$ is simply $1/x$. This formula is the basis for integrals involving $1/x$.

General Logarithm

&frac;d;a(x)] = &frac;1;

To differentiate a logarithm with a base other than $e$, you can use the change of base formula or memorize this result involving $\ln(a)$ in the denominator.

Differentiation Formulas

1. Basic Differentiation Rules

2. Product and Quotient Rules

3. The Chain Rule

4. Trigonometric Derivatives

5. Exponential and Logarithmic Derivatives

Basic Differentiation Formulas and Reference File Download Link

Differentiation Formulas and Reference File Download Link

Finite Difference Formulas For Numerical Differentiation and Reference File Download Link

Differentiation And Integration Formulas and Reference File Download Link

Primary Inputs, Formulas And Guidelines and Reference File Download Link

1. Basic Differentiation Rules

2. Product and Quotient Rules

3. The Chain Rule

4. Trigonometric Derivatives

5. Exponential and Logarithmic Derivatives

Basic Differentiation Formulas and Reference File Download Link

Differentiation Formulas and Reference File Download Link

Finite Difference Formulas For Numerical Differentiation and Reference File Download Link

Differentiation And Integration Formulas and Reference File Download Link

**Primary Inputs, Formulas And Guidelines** and Reference File Download Link

Primary Inputs, Formulas And Guidelines and Reference File Download Link