Find the derivative of the function given by f(x)=sin(x2).
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Step-by-Step Solution
Step 1: Identify the function and rule
The given function is f(x)=sin(x2). This is a composite function, meaning it's a function within a function. To differentiate such functions, we need to use the chain rule.
Step 2: Apply the Chain Rule
The chain rule states that the derivative of a composite function f(g(x)) is the derivative of the outer function f evaluated at the inner function g(x), multiplied by the derivative of the inner function g(x). In our case, the outer function is sin(u) and the inner function is u=x2.
Step 3: Differentiate the outer function
First, we differentiate the outer function, sin(u), with respect to u. The derivative of sin(u) is cos(u).
Step 4: Differentiate the inner function
Next, we differentiate the inner function, x2, with respect to x. Using the power rule, the derivative of x2 is 2x.
Step 5: Combine the derivatives
Finally, we combine the results from differentiating the outer and inner functions according to the chain rule. We substitute u=x2 back into the derivative of the outer function and multiply it by the derivative of the inner function.
Step 6: Simplify the expression
Rearranging the terms gives us the final simplified derivative of the function.