Question

Срочно пожалуйста помогите : Найти производную функцую : y=lncosx^-4

Answer 1

$y=ln(cosx^{-4})\; \; ,\; \; \; y=ln(cos\frac{1}{x^4})\\\\y'=\frac{1}{cos(x^{-4})}\cdot (-sin(x^{-4}))\cdot (-4x^{-5})=\frac{4\, sin\frac{1}{x^4}}{x^5\cdot cos\frac{1}{x^4}}= \frac{4}{x^5}\cdot tg \frac{1}{x^4}$

Answer 2

Finding the Derivative of y = ln(cos(x^-4))

To find the derivative of the function y = ln(cos(x^-4)), we can use the chain rule and the derivative rules for logarithmic and trigonometric functions.

The chain rule states that if we have a composite function, such as ln(f(x)), then the derivative is given by the derivative of the outer function multiplied by the derivative of the inner function.

Let's break down the steps to find the derivative of y = ln(cos(x^-4)):

Step 1: Apply the chain rule The outer function is ln(u), where u = cos(x^-4). The derivative of ln(u) with respect to x is 1/u multiplied by the derivative of u with respect to x.

Step 2: Find the derivative of the inner function The inner function is u = cos(x^-4). To find its derivative, we need to apply the chain rule again. The derivative of cos(u) with respect to u is -sin(u), and the derivative of u with respect to x is given by the power rule.

Step 3: Substitute the derivative of the inner function into the chain rule Now, we substitute the derivative of the inner function into the chain rule expression from step 1.

Putting it all together, the derivative of y = ln(cos(x^-4)) is:

dy/dx = (1/u) * du/dx = (1/cos(x^-4)) * (-sin(x^-4)) * d/dx(x^-4) = -sin(x^-4) * (1/cos(x^-4)) * d/dx(x^-4)

To simplify further, we can rewrite x^-4 as 1/x^4 and apply the power rule to find the derivative of x^-4:

d/dx(x^-4) = -4/x^5

Substituting this back into the expression, we have:

dy/dx = -sin(x^-4) * (1/cos(x^-4)) * (-4/x^5) = 4sin(x^-4) / (x^5 * cos(x^-4))

So, the derivative of y = ln(cos(x^-4)) is:

dy/dx = 4sin(x^-4) / (x^5 * cos(x^-4))

Please note that the above derivative is derived based on the given function, and it's always a good practice to double-check the result for accuracy.

I hope this helps! Let me know if you have any further questions.