Question

Найти производные функций : 1) -6arcsin(4x)-3sin(4x); 2) -10arccos(6x)e^10x;

Answer 1

1) 

2)$\frac{60}{ \sqrt{1-36* x^{2} } } e ^{10x} -100arccos(6x)e ^{10x}$

Answer 2

1) -6arcsin(4x) - 3sin(4x)

To find the derivative of the function -6arcsin(4x) - 3sin(4x), we can use the chain rule and the derivative rules for arcsin and sin functions.

The derivative of the arcsin function is given by:

d(arcsin(u))/dx = 1/sqrt(1-u^2) * du/dx

The derivative of the sin function is given by:

d(sin(u))/dx = cos(u) * du/dx

Let's find the derivative step by step:

Step 1: Find the derivative of -6arcsin(4x) - Applying the chain rule, we have: d(-6arcsin(4x))/dx = -6 * (1/sqrt(1-(4x)^2)) * d(4x)/dx

Step 2: Find the derivative of -3sin(4x) - Applying the chain rule, we have: d(-3sin(4x))/dx = -3 * cos(4x) * d(4x)/dx

Step 3: Simplify the derivatives obtained in steps 1 and 2 - Simplifying the derivatives, we have: d(-6arcsin(4x))/dx = -6 * (1/sqrt(1-(4x)^2)) * 4 d(-3sin(4x))/dx = -3 * cos(4x) * 4

Step 4: Combine the derivatives - Combining the derivatives, we have: d(-6arcsin(4x))/dx - d(-3sin(4x))/dx = -6 * (1/sqrt(1-(4x)^2)) * 4 - 3 * cos(4x) * 4

Therefore, the derivative of the function -6arcsin(4x) - 3sin(4x) is: -6 * (1/sqrt(1-(4x)^2)) * 4 - 3 * cos(4x) * 4

2) -10arccos(6x)e^10x

To find the derivative of the function -10arccos(6x)e^10x, we can use the product rule and the derivative rule for arccos and exponential functions.

The derivative of the arccos function is given by:

d(arccos(u))/dx = -1/sqrt(1-u^2) * du/dx

The derivative of the exponential function is given by:

d(e^u)/dx = e^u * du/dx

Let's find the derivative step by step:

Step 1: Find the derivative of -10arccos(6x) - Applying the chain rule, we have: d(-10arccos(6x))/dx = -10 * (-1/sqrt(1-(6x)^2)) * d(6x)/dx

Step 2: Find the derivative of e^10x - Applying the exponential function derivative rule, we have: d(e^10x)/dx = e^10x * d(10x)/dx

Step 3: Simplify the derivatives obtained in steps 1 and 2 - Simplifying the derivatives, we have: d(-10arccos(6x))/dx = 10 * (1/sqrt(1-(6x)^2)) * 6 d(e^10x)/dx = e^10x * 10

Step 4: Combine the derivatives - Combining the derivatives, we have: d(-10arccos(6x))/dx * e^10x + -10arccos(6x) * d(e^10x)/dx

Therefore, the derivative of the function -10arccos(6x)e^10x is: 10 * (1/sqrt(1-(6x)^2)) * 6 * e^10x + -10arccos(6x) * e^10x

Please note that the above derivatives are calculated based on the given functions and derivative rules.