Mathematics RU

Derivatives

Derivatives of powers and polynomials, products and quotients, composite functions — from x² to (x² − 2)⁵ and ln x / x. Endless new functions, each solved.

How to solve it

A derivative shows how fast a function changes. For simple functions it comes from a table; for combined ones, from three rules: the derivative of a sum is the sum of the derivatives, products and quotients have their own formulas, and for a composite function the derivative of the outer function is multiplied by the derivative of the inner one.

Step by step

Write roots and fractions as powers: $\frac{6}{x^2} = 6x^{-2}$, $\sqrt{x} = x^{1/2}$. Then $(x^n)' = nx^{n-1}$ applies.
A sum: differentiate each term, take constant factors out.
A product: $(uv)' = u'v + uv'$. A quotient: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.
A composite function: the derivative of the outer function (at the same argument) times the derivative of the inner one.
Simplify the answer.

The outer function.

The inner function, the outer function's argument.

The derivative of the inner function, the factor most often forgotten.

Example: $\bigl((x^2 - 2)^5\bigr)' = 5(x^2 - 2)^4 \cdot 2x = 10x(x^2 - 2)^4$.

Common mistakes

Forgetting the factor from the inner function: $(\sin 3x)' = 3\cos 3x$, not $\cos 3x$.
Differentiating a product as the product of derivatives: $(uv)' \ne u'v'$.
Swapping the order in the quotient rule's numerator: it is $u'v - uv'$.
Errors with negative exponents: $(x^{-2})' = -2x^{-3}$.