Since the function f(x) is a quotient of two separate functions, sin(x) + cos(x) and e^x, use the quotient rule to take the derivative.

Recall the quotient rule for two functions u and v:

Let's define u and v for this function as follows:

Take the derivative of u and v:

Now that we've defined u, u', v, and v' we can substitute them into the quotient rule formula.

Factor out e^x from to the top and bottom and simplify into the final form:

Since the function f(x) is a product of two functions, 5x^4 and sec(x), use the product rule to take this derivative.

Recall the product rule where u and v are two separate functions:

让我们来定义u和v如下:

Take the derivative of u and v:

Now that we've found values of u, u', v, and v' we can substitute them into the product rule formula for the final answer.

First notice that the function is a product of two functionsand. Apply the product rule:

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The second term will require the chain rule. Recall that the derivative for a radical function-of-a-functionis given by:

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The derivative can be computed using the quotient rule:

The function is a product of the functionsand, so apply the product rule:

The derivative in the first term isThe derivative of the second term requires the use of the chain rule.

The derivative in the second term required the use of the chain rule. First we write the derivative ofwith respect to the function, which is just. Then we multiply by the derivative ofwith respect to, which is just.

Therefore,

Given that there are 2 terms in the numerator and only one in the denominator, one can split up the equation into 2 separate derivatives:

.

Now we simplify these, and proceed to solve:

Because this problem contains two functions multiplied together that can't be simplified any further, it calls for the product rule, which states that

By applying this rule to the equation

we get

Because we are dealing with a quotient that cannot be simplified, we use the quotient rule, which states that if

,

.

By observing the given equation

,

we can see that

and

.

Therefore, the derivative is

.

Because we are differentiating a quotient that cannot be simplified, we must use the quotient rule, which states that if

,

then

.

By observing the given equation,

,

we see that in this case,

and

.

Given this information, the quotient rule tells us that

.

This problem is a quotient rule inside of a chain rule. First, let's look at the chain rule:

.

Given this, we can deduce that since

,

and

.

By plugging these into the chain rule formula, we get

To find the derivative of the

second term, we must use the quotient rule, which states that the the derivative

of a quotient is ((denominator)(derivative of numerator)-(numerator)(derivative

of denominator))/(denominator squared). Using this rules we find that

.

By plugging this back in, we find the final derivative to be