Question
Find f′(x) if f(x)=(cosx)^x .
Answers
GPT 3.5
To find f'(x), we can use the chain rule.
Let u = cos(x)
f(x) = u^x
Using the chain rule:
f'(x) = d/dx [u^x] = d/du [u^x] * du/dx
To find d/du [u^x], we can use the power rule for differentiation:
d/du [u^x] = x * u^(x-1)
To find du/dx, we can use the derivative of the cosine function:
du/dx = -sin(x)
Putting it all together:
f'(x) = x * (cos(x))^(x-1) * (-sin(x))
Simplifying:
f'(x) = -x * (cos(x))^(x-1) * sin(x)
Therefore, f'(x) = -x * (cos(x))^(x-1) * sin(x).
Let u = cos(x)
f(x) = u^x
Using the chain rule:
f'(x) = d/dx [u^x] = d/du [u^x] * du/dx
To find d/du [u^x], we can use the power rule for differentiation:
d/du [u^x] = x * u^(x-1)
To find du/dx, we can use the derivative of the cosine function:
du/dx = -sin(x)
Putting it all together:
f'(x) = x * (cos(x))^(x-1) * (-sin(x))
Simplifying:
f'(x) = -x * (cos(x))^(x-1) * sin(x)
Therefore, f'(x) = -x * (cos(x))^(x-1) * sin(x).