# What is the equation of the line normal to # f(x)=ln(x+xe^(3x))# at # x=2#?

We can rewrite the equation as follows (and consequently avoid needing to use the product rule):

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

To find the equation of the line normal to the function f(x) = ln(x + xe^(3x)) at x = 2, we need to determine the slope of the tangent line at x = 2 and then find the negative reciprocal of that slope to obtain the slope of the normal line.

To find the slope of the tangent line, we can use the derivative of the function. Taking the derivative of f(x) with respect to x, we get:

f'(x) = (1 + e^(3x))/(x + xe^(3x))

Evaluating f'(2), we find:

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

Simplifying this expression, we have:

f'(2) = (1 + e^6)/(2 + 2e^6)

Now, to find the slope of the normal line, we take the negative reciprocal of f'(2):

slope of normal line = -1/f'(2)

Finally, we have the equation of the line normal to f(x) at x = 2:

y - f(2) = slope of normal line * (x - 2)

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

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the equation of the line tangent to #f(x)=x^2-x # at #x=2#?
- Consider the curve defined by the equation #y+cosy=x+1# for #0≤y≤2pi#, how do you find dy/dx in terms of y and write an equation for each vertical tangent to the curve?
- What is the equation of the line that is normal to #f(x)= (x+2)^2-5x+2 # at # x=3 #?
- What is the equation of the line normal to #f(x)=- tan(2pix -2)# at #x=3#?
- What is the equation of the line normal to # f(x)=(x-1)(x+2) # at # x=0#?

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