Differentiating | Mathematico

Differentiating $e^{x}$

The number $e$ is, as we have mentioned before, a very important number. It is important because

y = e^{x} \Rightarrow \frac{d y}{d x} = e^{x}

We sometimes say that $e^{x}$ differentiates to itself.

This means that the gradient of the curve $y = e^{x}$ is equal to $e^{x},$ as you can see by moving the below point around:

Differentiating $e^{k x}$

We can also differentiate functions of the form

y = e^{k x}

where $k$ is some number.

f (x) = e^{x}

then

f (k x) = e^{k x}

is an enlargement in the $x$ direction of scale factor $\frac{1}{k} .$ This makes the gradient steeper by a scale factor of $k,$ as you can see below:

This suggests that

y = e^{k x} \Rightarrow \frac{d y}{d x} = k e^{k x}

We're going to assume this is true for now - we'll be able to prove in a couple of chapters' time when we study the chain rule.

Let

f (x) = (e^{\frac{x}{2}} - 3 e^{- x}) e^{x}

and

g (x) = \frac{e^{x} + 2 e^{3 x}}{e^{5 x}}

Find $f^{'} (x)$ and $g^{'} (x)$

Differentiating ex ​

Differentiating ekx ​

Differentiating $e^{x}$

Differentiating $e^{k x}$