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Answer is given, explain the full process including final steps

Answer is given, explain the full process including final steps-example-1

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Given the curve:


y=(18)/(x^2+2)

You need to find the equation for the tangent to that curve at the point:


(1,6)

Then, you need to follow these steps:

1. Derivate the function given in the exercise using these Derivative Rules:


(d)/(dx)((1)/(u(x)))=-(u^(\prime)(x))/(u(x)^2)
(d)/(dx)(x^n)=nx^(n-1)
(d)/(dx)(k)=0

Where "k" is a constant.

Then, you get:


=18\cdot((x^2+2)^(\prime))/((x^2+2)^2)
=-18\cdot((x^2+2)^(\prime))/((x^2+2)^2)
=-18\cdot(2x)/((x^2+2)^2)
y^(\prime)=-(36x)/((x^2+2)^2)

2. Substitute this value of the x-coordinate of the given point into the derivated function and then evaluate, in order to find the slope of the line:


m=-(36(1))/((1^2+2)^2)=-(36)/(9)=-4

4. The Point-Slope Form of the equation of a line is:


y-y_1=m(x-x_1)

Where "m" is the slope of the line and this point is on the line:


(x_1,y_1)

In this case:


\begin{gathered} m=-4 \\ x_1=1 \\ y_1=6 \end{gathered}

Therefore, you can substitute values:


y-6=-4(x-1)

5. Convert the equation from Point-Slope Form to Slope-Intercept Form.

The equation of a line in Slope-Intercept Form is:


y=mx+b

Where "m" is the slope and "b" is the y-intercept.

Then, by solving for "y", you get:


y-6=(-4)(x)-(-4)(1)


y-6=-4x+4
y=-4x+4+6
y=-4x+10

Hence, the answer is:

1. Derivate the function.

2. Find the slope of the tangent line by substituting the x-coordinate of the given point into the function derivated.

3. Write the equation of the tangent line in Point-Slope Form using the given point and the slope.

4. Rewrite the equation in Slope-Intercept Form by solving for "y".

Equation for the tangent of the curve:


y=-4x+10

User JoshOrndorff
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