# What is the equation of the tangent line of #f(x)=sqrt((x-1)^3e^(2x) # at #x=2#?

To solve this problem as presented, we must do the following:

(1)

(2)

Recall that the product rule states for

Additionally, the chain rule states that given

Here, we have the following:

where we take

By signing up, you agree to our Terms of Service and Privacy Policy

The equation of the tangent line is

The equation of a line in the form

Let's begin by getting the y-value of the point we want the line to go through:

Then we use the chain rule:

solving for the slope at the point of interest:

finally the equation of the tangent line is

which we can graph to check our solution:

graph{(5(e^2)/2x-4e^2-y)(sqrt((x-1)^3e^(2x))-y)=0 [1 3 -20 20]}

By signing up, you agree to our Terms of Service and Privacy Policy

To find the equation of the tangent line of f(x) = sqrt((x-1)^3e^(2x)) at x = 2, we need to find the slope of the tangent line and a point on the line.

First, we find the derivative of f(x) using the chain rule and product rule. The derivative is given by:

f'(x) = (3(x-1)^2e^(2x) + 2(x-1)^3e^(2x))/(2sqrt((x-1)^3e^(2x)))

Next, we substitute x = 2 into f'(x) to find the slope of the tangent line:

f'(2) = (3(2-1)^2e^(2*2) + 2(2-1)^3e^(2*2))/(2sqrt((2-1)^3e^(2*2)))

Simplifying this expression gives us the slope of the tangent line.

Finally, we use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, to find the equation of the tangent line. We substitute the values of x1, y1, and m into the equation and simplify to obtain the final equation of the tangent line.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you find the x-coordinates of all points on the curve #y=sin2x-2sinx# at which the tangent line is horizontal?
- How do you find the equation of the tangent line to the curve #y = (3x-1)(2x+4)# at the point of (0,-4) ?
- What is the equation of the normal line of #f(x)= cscx# at #x = pi/8#?
- How do you find the equations of the tangent lines to the curve #y= (x-1)/(x+1)# that are parallel to the line x-2y=5?
- How do you find instantaneous velocity from a position vs. time graph?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7