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lucphysics
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Homework Statement
f(x)= (sin x)^(sin x)
Homework Equations
The Attempt at a Solution
Taking logarithm on both sides I get:
ln y = ln ((sin x)^(sin x))
That's a good start. Can you rewrite the right side by using a property of the natural log function?lucphysics said:Homework Statement
f(x)= (sin x)^(sin x)
Homework Equations
The Attempt at a Solution
Taking logarithm on both sides I get:
ln y = ln ((sin x)^(sin x))
The derivative of (sin x)^sin x is cos x * (sin x)^sin x * (ln(sin x) + cos x * ln(sin x)). This can also be written as (sin x)^sin x * (cos x * ln(sin x) + ln(cos x))
To find the derivative of (sin x)^sin x, you can use the chain rule. First, rewrite the function as e^(sin x * ln(sin x)). Then, take the derivative of the exponent, sin x * ln(sin x), using the product rule and the derivative of sin x. Finally, multiply the derivative of the exponent by the original function, e^(sin x * ln(sin x)).
Yes, you can use the power rule to simplify the derivative of (sin x)^sin x to cos x * (sin x)^sin x * (1 + ln(sin x)).
First, rewrite the function as e^(sin x * ln(sin x)). Then, take the derivative of the exponent using the product rule: (ln(sin x) + cos x * ln(sin x)). Next, use the chain rule to find the derivative of sin x, which is cos x. Finally, multiply the derivative of the exponent by the original function, e^(sin x * ln(sin x)).
Yes, you can also use logarithmic differentiation to find the derivative of (sin x)^sin x. This involves taking the natural log of both sides of the original function, then using properties of logarithms to simplify the expression. Finally, take the derivative of both sides and solve for the derivative of (sin x)^sin x.