Derivatives of e^x, ln x, sin x and cos x

Besides the power rule you need four basic derivatives by heart. Everything else (product, quotient and chain rule) builds on them.

Rules

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3e^x, ln, sin, cos

Task

Differentiate:

f (x) = 5 e^{x} - 3 ln (x) + 5 sin (x)

$f^{'} (x) =$

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Differentiate: $f (x) = 5 e^{x} - 4 ln (x) - 2 sin (x)$

Worked solution

1.Basic derivatives: $(e^{x})^{'} = e^{x}$ , $(ln x)^{'} = \frac{1}{x}$ , $(sin x)^{'} = cos x$ , $(cos x)^{'} = - sin x$ .
2.Constant factors stay (constant multiple rule).
3. $f^{'} (x) = 5 e^{x} - \frac{4}{x} - 2 cos (x)$

Typical mistake: Writing $(ln x)^{'} = ln x$ or $(sin x)^{'} = - cos x$ (the minus belongs to $(cos x)^{'}$ ).

Common questions

The exponential function is exactly the function whose slope at every point equals its value. That is why e is the natural base.

Differentiating goes round in a circle: sin → cos → −sin → −cos → sin. Only cos picks up the minus sign.