ind an equation of the tangent line to the curve at the given point.

y = sin 6x + sin2 6x, (0, 0)

at any x the slope is y' = 6 cos6x + 12 sin6x cos6x

y'(0) = 6

so, the line through (0,0) with slope 6 is y=6x

Sure! Let's find the equation of the tangent line at the point (0, 0).

To find the slope of the tangent line, we need to find the derivative of the curve. So, let's differentiate y = sin 6x + sin² 6x with respect to x.

Using the chain rule, the derivative of sin 6x is 6 cos 6x, and the derivative of sin² 6x is 2 sin 6x multiplied by the derivative of 6x, which is 6.

Putting it all together, the derivative of y is:
dy/dx = 6 cos 6x + 12 sin 6x cos 6x.

Now that we have the derivative, let's substitute the given x-coordinate, which is 0, into the derivative: dy/dx at x = 0.

dy/dx at x = 0 = 6 cos (6 * 0) + 12 sin (6 * 0) cos (6 * 0)
dy/dx at x = 0 = 6 cos 0 + 12 sin 0 cos 0
dy/dx at x = 0 = 6 * 1 + 12 * 0 * 1
dy/dx at x = 0 = 6

So, the slope of the tangent line at the point (0, 0) is 6.

Now, using the point-slope form of the line, we can write the equation of the tangent line:

y - y₁ = m(x - x₁), where (x₁, y₁) is (0, 0) and m is the slope we found.

Simplifying, we get:
y - 0 = 6(x - 0)
y = 6x

Therefore, the equation of the tangent line to the curve y = sin 6x + sin² 6x at the point (0, 0) is y = 6x.

To find the equation of the tangent line to the curve at the given point (0, 0) for the equation y = sin(6x) + sin^2(6x), we need to find the derivative of the equation and evaluate it at x = 0.

Step 1: Find the derivative of y with respect to x using the chain rule.

y = sin(6x) + sin^2(6x)

To find the derivative, we will differentiate each term separately using the chain rule.

Derivative of sin(6x): (cos(6x) * 6)
Derivative of sin^2(6x): 2(sin(6x)) * (cos(6x) * 6)

y' = (cos(6x) * 6) + 2(sin(6x)) * (cos(6x) * 6)

Step 2: Simplify the equation.

y' = 6cos(6x) + 12sin(6x) * cos(6x)

Step 3: Evaluate the derivative at x = 0 to find the slope of the tangent line at the given point (0, 0).

slope = y'(0)
= 6cos(6(0)) + 12sin(6(0)) * cos(6(0))
= 6cos(0) + 12sin(0) * cos(0)
= 6(1) + 12(0) * 1
= 6

The slope of the tangent line at the point (0, 0) is 6.

Step 4: Use the point-slope formula to find the equation of the tangent line.

Using the formula: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the curve.

Plug in the values:
m = 6
x1 = 0
y1 = 0

y - 0 = 6(x - 0)
y = 6x

Therefore, the equation of the tangent line to the curve y = sin(6x) + sin^2(6x) at the point (0, 0) is y = 6x.

To find the equation of the tangent line to the curve at the given point, we need to find the derivative of the function and evaluate it at the given point.

Step 1: Find the derivative of the function y = sin 6x + sin^2 6x.
To do this, we need to apply the chain rule and power rule. Let's start with the first term.

The derivative of sin u, where u is a function of x, is cos u multiplied by the derivative of u with respect to x:
d/dx(sin u) = cos u * du/dx

In this case, u = 6x, so the derivative of sin 6x is cos 6x * d/dx(6x).
Using the power rule for the second term, we have:
d/dx(sin^2 6x) = 2 * sin 6x * cos 6x * d/dx(6x)

Combining these results, the derivative of y with respect to x is:
dy/dx = cos 6x * 6 + 2 * sin 6x * cos 6x * 6

Simplifying this expression, we have:
dy/dx = 6(cos 6x + 2sin 6x * cos 6x)

Step 2: Evaluate the derivative at the given point (0, 0).
Substitute x = 0 into the derivative:
dy/dx = 6(cos 0 + 2sin 0 * cos 0)
dy/dx = 6(1 + 0)
dy/dx = 6

The derivative of y with respect to x at x = 0 is 6.

Step 3: Use the slope-intercept form to find the equation of the tangent line.
The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

We already know the slope (m) of the tangent line, which is 6.

To find the y-intercept (b), substitute the coordinates of the given point (0, 0) into the equation:
0 = 6(0) + b
0 = b

Hence, the y-intercept is 0.

Therefore, the equation of the tangent line to the curve y = sin 6x + sin^2 6x at the point (0, 0) is:
y = 6x