Study Notes: Essential Differentiation Rules

The Constant Rule

The derivative of a constant function is 0.

Power Rule

If f(x) = x^n, then f'(x) = nx^{n-1}.

Constant Multiple Rule

If c is a constant and f is a differentiable function, then d/dx[cf(x)] = c * d/dx[f(x)].

Sum Rule

d/dx[f(x) + g(x)] = f'(x) + g'(x).

Difference Rule

d/dx[f(x) - g(x)] = f'(x) - g'(x).

Rewrite → Differentiate → Simplify

Systematic approach to obtain derivatives.

Derivative of e^x

The derivative of f(x) = e^x is f'(x) = e^x.

Derivative of ln x

The derivative of f(x) = ln x is f'(x) = 1/x, for x > 0.

Derivative of a^x

The derivative of f(x) = a^x is f'(x) = a^x ln(a).

Derivative of log_a x

The derivative of f(x) = log_a x is f'(x) = 1/(x ln(a)).

Derivative of sin x

The derivative of f(x) = sin x is f'(x) = cos x.

Derivative of cos x

The derivative of f(x) = cos x is f'(x) = -sin x.

Derivative of tan x

The derivative of f(x) = tan x is f'(x) = sec^2 x.

Derivative of cot x

The derivative of f(x) = cot x is f'(x) = -csc^2 x.

Derivative of sec x

The derivative of f(x) = sec x is f'(x) = sec x tan x.

Derivative of csc x

The derivative of f(x) = csc x is f'(x) = -csc x cot x.

Confusion Between Power Rule and Exponential Rules

The Power Rule applies to variable bases; exponential rules apply to constant bases.

Failing to Rewrite Radicals

Always rewrite radicals as exponents before differentiating.

Sign Errors with Trig Functions

Trigonometric functions starting with 'c' have negative derivatives.

Notation Sloppiness

Correctly use notation to distinguish between function and its derivative.

Evaluating Before Differentiating

Find the general derivative first, then substitute x-values.

Radicals and Rational Expressions

Use the Power Rule by rewriting them as exponents first.

Linearity of Derivatives

Differentiation is a linear operator; you can split and factor derivatives.

Worked Example: y = 3x^4 - 5√x + 2/x + 7

Derivative is found by rewriting, differentiating, and simplifying.

Mnemonic for Trigonometric Derivatives

'PSST' helps remember the relationships: Tangent with Secant, Cotangent with Cosecant.

Common Pitfalls in Differentiation

Includes confusing power and exponential rules and failing to rewrite radicals.

Derivative of 2^x

f'(x) = 2^x ln(2), not x * 2^(x-1).