Finding derivative of f(X) = sin(cos(x^5))

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In summary, a derivative is the slope of the tangent line to a function at a specific point and it helps us understand how the function is changing. To find the derivative of a function, we use a process called differentiation. The chain rule is a formula used to find the derivative of a function composed of two or more functions. To apply the chain rule, we find the derivative of the outer function and multiply it by the derivative of the inner function. Finding derivatives is important because they have practical applications in various fields and help us understand how quantities change over time.
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i3uddha
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alright so i was given this problem:
f(X) = sin(cos(x^5))
and it tell me to find the first derivative of it.
would i first solve the derivative of (cos(x^5)) first and then that answer with sin?
or would i use the chain rule?
thanks in advanced.
 
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  • #2
Chain rule.

Start by putting [itex]u=cos(x^2)[/itex] and [itex]y = sin u.[/itex]
 
  • #3
Chain rule.

Start by putting [itex]u=cos(x^2)[/itex] and [itex]y = sin u.[/itex]
 

FAQ: Finding derivative of f(X) = sin(cos(x^5))

What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point, and it helps us understand how the function is changing.

How do you find the derivative of a function?

To find the derivative of a function, we use a mathematical process called differentiation. This involves using rules and formulas to find the slope of the tangent line at a specific point on the function.

What is the chain rule in finding derivatives?

The chain rule is a formula used to find the derivative of a function that is composed of two or more functions. In other words, if our function is a composition of two or more functions, we use the chain rule to find its derivative.

How do you apply the chain rule to find the derivative of f(X) = sin(cos(x^5))?

To apply the chain rule in this case, we first find the derivative of the outer function (sin) and then multiply it by the derivative of the inner function (cos(x^5)). In this case, the derivative of sin(x) is cos(x) and the derivative of cos(x^5) is -5x^4sin(x^5). Therefore, the derivative of f(x) = sin(cos(x^5)) is cos(cos(x^5)) * (-5x^4sin(x^5)).

Why is finding derivatives important?

Derivatives have many practical applications in various fields such as physics, engineering, and economics. They help us understand how quantities are changing over time, and they are also used to optimize functions and solve real-world problems.

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