1. 8sin(cos(x^6)), find f'(x)
2. cos(sin(x^2)), find f'(x).
my answer was -40cos(sincos(x^5) right?
Go on :
calc101 dot com
When page be open click option : derivatives
When page be open in rectangle type:
8 sin [ cos [ x ^ 6 ] ]
and click option : DO IT
You will see solution step by step
Then in rectangle type :
cos [ sin [ x ^ 2 ] ]
and click option : DO IT
To find the derivative of a function, we can use the chain rule. The chain rule states that if we have a function g(x) nested inside another function f(x), the derivative of f(g(x)) with respect to x is the derivative of f(g(x)) multiplied by the derivative of g(x) with respect to x.
1. To find the derivative of f(x) = 8sin(cos(x^6)), we start by applying the chain rule. Let's define g(x) = cos(x^6). Then f(x) can be rewritten as f(g(x)) = 8sin(g(x)).
To find the derivative, f'(x), we need to find f'(g(x)), which represents the derivative of f with respect to g. We also need to find g'(x), which represents the derivative of g with respect to x.
Step 1: Find g'(x):
g(x) = cos(x^6)
Using the chain rule, we have g'(x) = -sin(x^6) * (6x^5)
g'(x) = -6x^5sin(x^6)
Step 2: Find f'(g(x)):
f(g(x)) = 8sin(g(x))
Using the chain rule, we have f'(g(x)) = 8cos(g(x)) * g'(x)
f'(g(x)) = 8cos(g(x)) * (-6x^5sin(x^6))
Step 3: Multiply f'(g(x)) by g'(x) to get f'(x):
f'(x) = f'(g(x)) * g'(x)
f'(x) = 8cos(g(x)) * (-6x^5sin(x^6)) * (-6x^5sin(x^6))
f'(x) = 48x^10sin^2(x^6)cos(g(x))
Therefore, f'(x) = 48x^10sin^2(x^6)cos(g(x)).
2. To find the derivative of f(x) = cos(sin(x^2)), we again apply the chain rule. Let's define g(x) = sin(x^2). Then f(x) can be rewritten as f(g(x)) = cos(g(x)).
Step 1: Find g'(x):
g(x) = sin(x^2)
Using the chain rule, we have g'(x) = cos(x^2) * (2x)
g'(x) = 2x*cos(x^2)
Step 2: Find f'(g(x)):
f(g(x)) = cos(g(x))
Using the chain rule, we have f'(g(x)) = -sin(g(x)) * g'(x)
f'(g(x)) = -sin(g(x)) * (2x*cos(x^2))
Step 3: Multiply f'(g(x)) by g'(x) to get f'(x):
f'(x) = f'(g(x)) * g'(x)
f'(x) = -sin(g(x)) * (2x*cos(x^2)) * (2x*cos(x^2))
f'(x) = -4x^2*sin(g(x))*cos^2(x^2))
Therefore, f'(x) = -4x^2*sin(g(x))*cos^2(x^2).