Question

А) F(x)=sin 5x *sin 3x + cos 5x*cos3x б) F (x) = sin 3x*cos3x

Answer 1

Первое sin(5x+3x) вроде бы правильно

Answer 2

Let's find the derivatives of the given functions step by step.

a) F(x) = sin(5x) * sin(3x) + cos(5x) * cos(3x)

To find the derivative of F(x) with respect to x, we can use the product rule and the trigonometric derivative identities.

Product Rule: d(uv)/dx = u * dv/dx + v * du/dx Trigonometric Derivative Identities: d(sin(x))/dx = cos(x) and d(cos(x))/dx = -sin(x)

Let's calculate the derivative:

F'(x) = (sin(5x) * d(sin(3x))/dx) + (cos(5x) * d(cos(3x))/dx) + (d(sin(5x))/dx * sin(3x)) + (d(cos(5x))/dx * cos(3x))

F'(x) = (sin(5x) * cos(3x)) + (cos(5x) * (-sin(3x))) + (cos(5x) * sin(3x)) + (-sin(5x) * cos(3x))

Now, we can combine like terms:

F'(x) = sin(5x) * cos(3x) - sin(3x) * cos(5x) + cos(5x) * sin(3x) - sin(5x) * cos(3x)

Finally, we can simplify it further:

F'(x) = sin(5x) * cos(3x) - sin(3x) * cos(5x)

b) F(x) = sin(3x) * cos(3x)

To find the derivative of F(x) with respect to x, we'll use the product rule again.

F'(x) = (sin(3x) * d(cos(3x))/dx) + (cos(3x) * d(sin(3x))/dx)

Using the trigonometric derivative identities: d(sin(x))/dx = cos(x) and d(cos(x))/dx = -sin(x)

F'(x) = (sin(3x) * (-sin(3x))) + (cos(3x) * cos(3x))

F'(x) = -sin^2(3x) + cos^2(3x)

Finally, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1:

F'(x) = 1 - 2sin^2(3x)

These are the derivatives of the given functions. If you have any further questions or need additional explanations, feel free to ask!