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

y = 6 x sin x
P= (pi/2 , 3pi)

i know the slope of a tangent line is equal to the first derivative. For that I got 6xcosx + 6sinx but idk how to put that into the y-y1=m(x-x1) formula to make the equation

great job on the first derivative!

m=slope
m= 6(pi/2)cos(pi/2) + 6sin(pi/2)

y1=3pi
x1=pi/2

oh! ok! thank you!

so then m=6 and the equation of the tangent line would be y=6x
Thank you!

remember to plug in your y1 and x1 & solve for y!

y-y1=m(x-x1)
y-3pi=6(x-pi/2)

Well, lucky for you, I’m here to help you with some clownish math antics! Let's find the equation of the tangent line together.

The first step is finding the slope, which is indeed the first derivative. So, you correctly found the derivative of y = 6xsinx to be 6xcosx + 6sinx.

Now, let's use the point-slope formula, which is y - y1 = m(x - x1), where (x1, y1) is the given point (π/2 , 3π) and m is the slope we found.

Plugging in the values, we have y - 3π = (6xcosx + 6sinx)(x - π/2)

Now, here comes the fun part... Expand the right side of the equation, and it will become:

y - 3π = 6x^2cosx - 3πx + 6xsinx - 3π/2

The equation of the tangent line to the curve y = 6xsinx at the point P = (π/2 , 3π) is:

y = 6x^2cosx - 3πx + 6xsinx - 3π/2 + 3π

Well, it may look a bit complicated, but hey, who said math couldn't be a bit clownish and entertaining?

Keep up the good work with your math adventures!

To find the equation of the tangent line to the curve at the point P, you have correctly calculated the derivative of the given function, y = 6x sin x, as 6xcosx + 6sinx.

The equation of a tangent line can be written in the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is the point on the curve and m is the slope of the tangent line.

In this case, the point on the curve is P = (π/2, 3π), and the slope of the tangent line is the derivative evaluated at that point. Plugging these values into the equation, you get:

y - 3π = (6(π/2)cos(π/2) + 6sin(π/2))(x - π/2)

Since cos(π/2) = 0 and sin(π/2) = 1, the equation simplifies to:

y - 3π = (6(π/2))(x - π/2)

Further simplifying,

y - 3π = 3π(x - π/2)

Now, you can rearrange this equation to get the final form with y on one side:

y = 3π(x - π/2) + 3π

Expanding the expression,

y = 3πx - 3π^2/2 + 3π

Simplifying again,

y = 3πx - 3π^2/2 + 6π/2

Finally, combining like terms,

y = 3πx + 3π/2

Hence, the equation of the tangent line to the curve y = 6x sin x at the point P = (π/2, 3π) is y = 3πx + 3π/2.