Find the derivative of the expression e x cos ⁡ x

Question

Find the derivative of the expression       e          x        cos  ⁡  x

Rebecca Benitez · Accepted Answer

In case you are not familiar with all the notations, there are two main ways to indicate the derivative of a function:

\frac{d}{d x}

where

x

is the "with respect to" variable
Just an apostrophe, like

f' (x)

, or simply

f'

In the example

\frac{d}{d x} e^{x} \cos x

above, the

\frac{d}{d x}

notation tells us that we’re looking for the derivative of

e^{x} \cos x

. The product rule, meanwhile, says that the derivative of

f g

is equal to the derivative of

f

multiplied by

g

, plus

f

multiplied by the derivative of

g

.
In order to use the derivative product rule, as with any rule in calculus, first we need to assign parts of our expression to the appropriate variables in the rule. In this case, they are

f

and

g

. Let

f = e^{x}

and

g = \cos x

.
Now, we can use the product rule to give:

\frac{d}{d x} e^{x} \cos x = (\frac{d}{d x} e^{x}) \cos x + e^{x} (\frac{d}{d x} \cos x)

Using other derivative rules that we will not go into detail here, we know that the derivative of

e^{x}

is

e^{x}

,and the derivative of

\cos x

is

- \sin x

.
We are done. We have found the derivative:

\frac{d}{d x} e^{x} \cos x = e^{x} \cos x - e^{x} \sin x

Find the derivative of the expression e^x cos x

Answered question

Answer & Explanation

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