how do not understand how to find a deriviative of this function?Do i use chain rule and product rule?
r= the square root of (theta*sin*theta)
To find the derivative of the function r = √(θ * sin θ), yes, you can use both the chain rule and the product rule.
Let's break it down step by step:
Step 1: Identify which part of the function requires the chain rule. In this case, it is the square root of (θ * sin θ).
Step 2: Apply the chain rule to the square root part. The chain rule states that if you have a function inside another function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is the square root function, and the inner function is (θ * sin θ).
Step 3: Compute the derivative of the square root part, which is 1/2 * (θ * sin θ)^(-1/2).
Step 4: Now, we need to find the derivative of the inner function (θ * sin θ) using the product rule.
Step 5: Apply the product rule: If you have two functions u and v, the derivative of the product uv is given by the formula (u * v') + (u' * v). In this case, u = θ and v = sin θ.
Step 6: Compute the derivatives of u and v. The derivative of θ is simply 1, since it is a variable. The derivative of sin θ is cos θ.
Step 7: Apply the product rule using the derivatives from step 6. (θ * cos θ) + (1 * sin θ).
Step 8: Multiply the result from step 3 with the result from step 7 to get the final derivative of r.
In summary, the derivative of r = √(θ * sin θ) using both the chain rule and product rule is:
dr/dθ = 1/2 * (θ * sin θ)^(-1/2) * [(θ * cos θ) + (sin θ)]
This is the general process to find the derivative of the given function.