# What is the equation of the tangent line of #f(x)=x^3-12x/(x^2+9x)-1# at #x=-3#?

Equation of tangent line would be

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

To find the equation of the tangent line of the function f(x) = (x^3 - 12x) / (x^2 + 9x) - 1 at x = -3, 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 quotient rule:

f'(x) = [(x^2 + 9x)(3x^2 - 12) - (x^3 - 12x)(2x + 9)] / (x^2 + 9x)^2

Next, we substitute x = -3 into f'(x) to find the slope of the tangent line:

f'(-3) = [((-3)^2 + 9(-3))(3(-3)^2 - 12) - ((-3)^3 - 12(-3))(2(-3) + 9)] / ((-3)^2 + 9(-3))^2

Simplifying the expression, we get:

f'(-3) = -6/9

Therefore, the slope of the tangent line at x = -3 is -6/9, which simplifies to -2/3.

To find a point on the tangent line, we substitute x = -3 into the original function f(x):

f(-3) = (-3^3 - 12(-3)) / (-3^2 + 9(-3)) - 1

Simplifying the expression, we get:

f(-3) = -27/6 - 1

f(-3) = -29/6

Therefore, a point on the tangent line is (-3, -29/6).

Using the point-slope form of a line, we can write the equation of the tangent line:

y - y1 = m(x - x1)

Substituting the values we found, we get:

y - (-29/6) = (-2/3)(x - (-3))

Simplifying the equation, we have:

y + 29/6 = (-2/3)(x + 3)

Multiplying through by 6 to eliminate the fractions, we get:

6y + 29 = -4(x + 3)

Finally, simplifying the equation, we have:

6y + 29 = -4x - 12

Therefore, the equation of the tangent line of f(x) = (x^3 - 12x) / (x^2 + 9x) - 1 at x = -3 is 4x + 6y + 41 = 0.

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 average value of the function # f(x)=(x-1)^2# on the interval #[1,5]#?
- What is the average rate of change of the function #f(x)=2x^2 -3x -1# on the interval [2, 2.1]?
- How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrt(4x-5)#?
- What is the equation of the tangent line of #f(x)=sqrt(x+1)-sqrt(x+2) # at #x=2#?
- How do you find the derivative of # e^(xy)=x/y#?

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