Question

Решение производных Y=(x ^6+x ^5-5/x) ^1/3 Y= sin^3*5x+e^7x Y=tg^3*4x

Answer 1

Лоогаыпдгвнкнкнкнкнан исргршррп

Answer 2

To find the derivatives of the given functions, we can use the rules of differentiation. Let's find the derivatives of each function step by step:

1. Y = (x^6 + x^5 - 5/x)^(1/3)

To differentiate this function, we'll use the chain rule, which states that if we have a function of the form (u)^n, the derivative is n * (u)^(n-1) * u', where u' is the derivative of u with respect to x.

Let u = (x^6 + x^5 - 5/x), and n = 1/3.

Now, let's find u':

u = (x^6 + x^5 - 5/x)

u' = d/dx(x^6) + d/dx(x^5) - d/dx(5/x) u' = 6x^5 + 5x^4 + 5/x^2

Now, we can find the derivative of Y using the chain rule:

d/dx(Y) = (1/3) * (x^6 + x^5 - 5/x)^(-2/3) * (6x^5 + 5x^4 + 5/x^2)

2. Y = sin^3(5x) + e^(7x)

To find the derivative of this function, we'll use the sum rule, which states that if we have a function of the form f(x) + g(x), then its derivative is f'(x) + g'(x).

Let's find the derivative of each term separately:

d/dx(sin^3(5x)):

Using the chain rule, let u = sin(5x), and n = 3.

u' = d/dx(sin(5x)) = 5cos(5x)

Now, applying the chain rule:

d/dx(sin^3(5x)) = 3(sin^2(5x)) * d/dx(sin(5x)) = 3(sin^2(5x)) * 5cos(5x) = 15sin^2(5x)cos(5x)

Next, d/dx(e^(7x)):

The derivative of e^(kx) with respect to x is k * e^(kx), where k is a constant.

d/dx(e^(7x)) = 7 * e^(7x)

Now, we can find the derivative of Y using the sum rule:

d/dx(Y) = d/dx(sin^3(5x)) + d/dx(e^(7x)) = 15sin^2(5x)cos(5x) + 7e^(7x)

3. Y = tan^3(4x)

To find the derivative of this function, we'll use the chain rule, similar to the first example.

Let u = tan(4x), and n = 3.

Now, let's find u':

u = tan(4x)

u' = d/dx(tan(4x)) = 4sec^2(4x)

Now, applying the chain rule:

d/dx(tan^3(4x)) = 3(tan^2(4x)) * d/dx(tan(4x)) = 3(tan^2(4x)) * 4sec^2(4x) = 12tan^2(4x)sec^2(4x)

So, the derivatives of the given functions are:

1. d/dx(Y) = (1/3) * (x^6 + x^5 - 5/x)^(-2/3) * (6x^5 + 5x^4 + 5/x^2)
2. d/dx(Y) = 15sin^2(5x)cos(5x) + 7e^(7x)
3. d/dx(Y) = 12tan^2(4x)sec^2(4x)