Let f(x) = 6sin(sin(x^5)). Find f'(x)
Chain this
if f(x)=g(h(x)) then
f'(x)=g'*h'..
f'=6cos(sin(x^5))*cos(x^5)*5x^4
check that.
To find the derivative of f(x), we can use the chain rule. Let's break down the process step by step:
Step 1: Identify the inner function.
The inner function in this case is g(x) = sin(x^5).
Step 2: Find the derivative of the inner function.
To find the derivative of g(x), we can use the chain rule again.
g'(x) = cos(x^5) * 5x^4.
Step 3: Identify the outer function.
The outer function in this case is f(x) = 6sin(g(x)).
Step 4: Apply the chain rule.
We know that the derivative of sin(x) is cos(x).
Applying the chain rule, we have:
f'(x) = 6cos(g(x)) * g'(x).
Step 5: Substitute the derivative of the inner function into the equation.
From step 2, we found that g'(x) = cos(x^5) * 5x^4.
Substituting it into the equation, we have:
f'(x) = 6cos(g(x)) * [cos(x^5) * 5x^4].
Therefore, the derivative of f(x) is given by f'(x) = 6cos(g(x)) * [cos(x^5) * 5x^4].
To find the derivative of f(x), which is denoted as f'(x), we can use the chain rule.
The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
In the given function f(x) = 6sin(sin(x^5)), we have a composition of functions: the outer function is sin and the inner function is sin(x^5).
To find f'(x), we need to take the derivative of the outer function with respect to the inner function (sin(x^5)), and then multiply it by the derivative of the inner function with respect to x (x^5).
Let's go step by step:
Step 1: Take the derivative of the outer function:
The derivative of sin(x) is cos(x). So, the derivative of sin(sin(x^5)) with respect to sin(x^5) is cos(sin(x^5)).
Step 2: Take the derivative of the inner function:
The derivative of x^5 with respect to x is 5x^4.
Step 3: Apply the chain rule:
Multiply the derivative of the outer function by the derivative of the inner function:
f'(x) = 6cos(sin(x^5)) * 5x^4.
So, the derivative of f(x) = 6sin(sin(x^5)) with respect to x is f'(x) = 6cos(sin(x^5)) * 5x^4.