- #1
delfam
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Homework Statement
y=cot^7(x^5)
Homework Equations
f(x)=f(g(x))
The Attempt at a Solution
u=(x^5)
y'=7(-csc^2)^6(x^5) * 5x^4
The Chain Rule is a mathematical rule that allows you to find the derivative of a composite function. It is used when a function is composed of two or more functions.
To solve for y' using the Chain Rule, first identify the inner and outer functions. In this case, the inner function is x^5 and the outer function is cot^7. Then, apply the Chain Rule formula: y' = f'(g(x)) * g'(x). In this case, f(x) = cot^7(x) and g(x) = x^5. Finally, plug in the derivative of f(x) and g(x) and simplify the expression.
The derivative of cot^7(x) is -7cot^6(x)csc^2(x).
To simplify the expression for y', start by plugging in the derivative of cot^7(x) and x^5. Then, use trigonometric identities and algebraic manipulations to simplify the expression. Finally, rewrite the expression using only cot(x) and csc(x) terms.
Yes, the Chain Rule can be applied to any composite function. It is a general rule for finding the derivative of a composite function and can be used for other trigonometric functions, exponential functions, and more.