find dy/dx of y=1/2 cot^7(-x^2)
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use the chain rule. let
v = -x^2
u = cot v
y = 1/2 u^7
y' = 7/2 u^6 u'
u' = -csc^2 v v'
v' = -2x
So,
dy/dx = 7/2 cot^6(-x^2) * -csc^2(-x^2) * -2x
= 7x cot^2(-x^2) csc^2(-x^2)
or, since
cot(-x) = -cotx
csc(-x) = -csc(x)
7x cot^2(x^2) csc^2(x^2)
To find the derivative of y = (1/2) cot^7(-x^2), we can use the chain rule and the derivative rules for cotangent.
Step 1: Rewrite the equation using trigonometric identities.
Recall that cot^2(x) = 1/tan^2(x), so we can rewrite cot^7(-x^2) as (1/tan^2(-x^2))^7.
Step 2: Apply the chain rule.
Let's differentiate the function term by term.
First, we differentiate (1/2) with respect to x. The derivative of a constant is always 0.
Next, let's differentiate (tan^2(-x^2))^7 with respect to x. To do this, we need to apply the chain rule.
Differentiating (tan^2(-x^2))^7 involves two steps:
a) Differentiate the outer function u^7 with respect to u, where u = tan^2(-x^2).
b) Differentiate the inner function u = tan^2(-x^2) with respect to x.
Let's tackle these steps one by one.
Step 3: Differentiate the outer function.
To differentiate u^7 with respect to u, we multiply the function by the derivative of the exponent, which is 7u^(7-1) = 7u^6.
Therefore, (tan^2(-x^2))^7 differentiates to 7(tan^2(-x^2))^6.
Step 4: Differentiate the inner function.
Let's differentiate u = tan^2(-x^2) with respect to x.
This requires us to apply the chain rule again.
Differentiating tan^2(-x^2) involves two steps:
a) Differentiate the outer function v^2 with respect to v, where v = tan(-x^2).
b) Differentiate the inner function v = tan(-x^2) with respect to x.
Step 5: Differentiate the outer function.
To differentiate v^2 with respect to v, we multiply the function by the derivative of the exponent, which is 2v^(2-1) = 2v.
Therefore, tan^2(-x^2) differentiates to 2tan(-x^2).
Step 6: Differentiate the inner function.
Differentiating v = tan(-x^2) with respect to x involves the chain rule again.
The derivative of tan(u) with respect to u is sec^2(u).
Therefore, differentiating v = tan(-x^2) with respect to x gives us d/dx[tan(-x^2)] = sec^2(-x^2).
Step 7: Find the final derivative.
Now, let's put all the pieces together.
d/dx[(1/2) cot^7(-x^2)] = 0 + 7(tan^2(-x^2))^6 * d/dx[tan^2(-x^2)] * d/dx[-x^2]
= 7(tan^2(-x^2))^6 * 2tan(-x^2) * sec^2(-x^2) * (-2x)
= -28x(tan^2(-x^2))^6tan(-x^2)sec^2(-x^2).
Therefore, dy/dx = -28x(tan^2(-x^2))^6tan(-x^2)sec^2(-x^2).