Derivatives

The table of derivatives and the rules: sum, product, quotient, the chain rule.

How to solve it

Derivatives

The derivative shows how fast a function changes. It is found not from the definition but from a table and a few rules: the sum, the product, the quotient and the chain rule.

Step by step

The table: $(x^n)' = nx^{n - 1}$, $(e^x)' = e^x$, $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\ln x)' = \frac{1}{x}$, $(\sqrt{x})' = \frac{1}{2\sqrt{x}}$.
The derivative of a sum is the sum of the derivatives; a constant factor comes out of the derivative.
The product: $(uv)' = u'v + uv'$. The quotient: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.
The chain rule: the derivative of the outer function times the derivative of the inner one: $(\sin 3x)' = 3\cos 3x$.

The first factor.

The second factor. Each is differentiated in turn while the other stays as it is.

Example: $(x^2 e^{x})' = 2x e^{x} + x^2 e^{x}$.

The inner function. Its derivative appears as a factor.

Example: $\left(e^{x^2}\right)' = e^{x^2} \cdot 2x$.

Common mistakes

Thinking that the derivative of a product is the product of the derivatives. It is not.
Forgetting the factor from the inner function: $(\cos 5x)' = -5\sin 5x$.
Getting the sign of the cosine's derivative wrong: $(\cos x)' = -\sin x$.