Since the function f(x) is a quotient of two separate functions, 10^x and cos(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 for the final answer.

The function f(x) is a sum of two functions, 3x^7 and -x^10*sin(x). When functions are added together, take the derivative of each one separately. First take the derivative of 3x^7:

The second part of the function, -x^10*sin(x) is a product of two functions so use the product rule to take the derivative. Recall the product rule where u and v are two separate functions:

Let's define u and v as follows:

Take the derivative of u and v:

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

For the final answer add (uv)' to the derivative of 3x^7.

Since the function f(x) is a quotient of two separate functions, cos(x) and x^2+1, 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 for the final answer.

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:

Let's define u and v as follows:

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 functions  and . 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-function  is given by:

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The derivative can be computed using the quotient rule: 

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

The derivative in the first term is  The 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 of  with respect to the function , which is just . Then we multiply by the derivative of  with 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