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moe darklight
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Homework Statement
y = (3x)^x
(find y' )
Homework Equations
y = a^x
y' = (a^x)(ln(a))
and the chain rule
The Attempt at a Solution
3((3x)^x)(ln(3x))
The Chain Rule is a rule in calculus that helps us find the derivative of a composite function. In simpler terms, it helps us find the rate of change of a function within another function.
In order to use the Chain Rule, we first need to identify the inner and outer functions. In this case, the inner function is 3x and the outer function is x. Then, we use the formula for the Chain Rule: dy/dx = (dy/du)(du/dx), where u is the inner function. In this case, the derivative of 3x is 3, and the derivative of x is 1. So, the final derivative is dy/dx = 3x^(x-1) * (ln3 + 1).
Sure! Let's say we have the function f(x) = (4x + 3)^2. In this case, the inner function is 4x + 3 and the outer function is x^2. Using the Chain Rule, we can find the derivative by first finding the derivative of the inner function (dy/du = 8x) and then multiplying it by the derivative of the outer function (du/dx = 2x). So, the final derivative is dy/dx = 8x * 2x = 16x^2.
The Chain Rule is important because it allows us to find the derivatives of more complex functions. Without it, we would only be able to find the derivatives of simple functions like polynomials and trigonometric functions. With the Chain Rule, we can handle functions within functions, making it an essential tool in calculus.
Yes, there are a few common mistakes to watch out for when using the Chain Rule. One is forgetting to apply the derivative of the outer function, which is often just x. Another is mixing up the order of the derivative when finding the derivative of the inner function. It's important to remember that the derivative of the inner function is multiplied by the derivative of the outer function, not the other way around.