# What is the equation of the normal line of #f(x)=x^2/(1+4x)# at #x=-1#?

Equation of normal at

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

To find the equation of the normal line of f(x) at x=-1, we need to determine the slope of the tangent line at x=-1 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 f(x). Taking the derivative of f(x)=x^2/(1+4x) with respect to x, we get:

f'(x) = (2x(1+4x) - x^2(4))/(1+4x)^2

Evaluating f'(-1), we substitute x=-1 into the derivative:

f'(-1) = (2(-1)(1+4(-1)) - (-1)^2(4))/(1+4(-1))^2

Simplifying the expression, we find:

f'(-1) = -6/25

The slope of the tangent line at x=-1 is -6/25.

To find the slope of the normal line, we take the negative reciprocal of the slope of the tangent line:

m_normal = -1/(f'(-1))

Substituting the value of f'(-1), we have:

m_normal = -1/(-6/25)

Simplifying, we get:

m_normal = 25/6

Now that we have the slope of the normal line, we can use the point-slope form of a line to find the equation. Since the point of interest is x=-1, we can use the coordinates (-1, f(-1)).

Substituting the values into the point-slope form, we have:

y - f(-1) = (25/6)(x - (-1))

Simplifying, we get:

y - f(-1) = (25/6)(x + 1)

This is the equation of the normal line of f(x)=x^2/(1+4x) at x=-1.

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 that is normal to #f(x)=-3x^2sinx # at # x=pi/3#?
- How do you find the equation of the tangent line to the graph #f(x)=ln(e^(x^2))# through point (-2,4)?
- How to find the equation of the tangent and the normal of this curve?
- How do you find an equation of the tangent line to the curve at the given point #y = 4 cos x# and #x=pi/6#?
- What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(t+pi),sin(2t-pi/4)) # at # t=(-pi)/3 #?

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