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Cacophony
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Homework Statement
How do I know for sure when to use the power rule instead of the chain rule and vice versa?
The power rule, also known as the exponent rule, is a mathematical principle that allows us to find the derivative of a function raised to a power. It states that if a function f(x) is raised to a constant power n, then its derivative is n times f(x) to the power of n-1.
To apply the power rule, we simply need to bring the exponent down in front of the function and subtract 1 from the original exponent. For example, if we have the function f(x) = x^3, its derivative would be f'(x) = 3x^(3-1) = 3x^2.
The chains rule, also known as the chain rule, is a mathematical principle that allows us to find the derivative of a composite function. It states that if a function is composed of two or more functions, then its derivative is equal to the derivative of the outer function multiplied by the derivative of the inner function.
To apply the chains rule, we first identify the inner function and the outer function. Then, we take the derivative of the outer function, leaving the inner function untouched, and multiply it by the derivative of the inner function. For example, if we have the function f(x) = (3x^2+1)^5, its derivative would be f'(x) = 5(3x^2+1)^4 * 6x = 30x(3x^2+1)^4.
Yes, the power rule and chains rule can be used together to find the derivative of a function that involves both powers and composite functions. We first apply the chains rule to find the derivative of the composite function, and then apply the power rule to find the derivative of the resulting function raised to a power.