Question

Find the equation of the tangent to the curve at the point where x = 3, givingthe answer in the form y= mx+C

Answer 1

y = 6x - 18

STEP - BY - STEP EXPLANATION

What to find?

• The derivative of y.

,

• The value of y when x=3

,

• The value of the intercept c.

,

• The equation of tangent to the curve.

Given:

x = 3

y = In(x² - 8)

To find the equation of the tangent to the curve at x=3, we need to follow the steps below:

Step 1

Find the derivative of y.

Given that y is an implicit function, we will differentiate using the implicit rule.

That is;

`(dy)/(dx)=(1)/((x^2-8))*2x`

Hence,

`(dy)/(dx)=(2x)/(x^2-8)`

Step 2

Find the value of the derivative in step two at x=3

We can obtain this by substituting x=3 in the above.

`(dy)/(dx)=(2(3))/(3^2-8)`
`=(6)/(9-8)=(6)/(1)=6`

Hence, dy/dx = 6

This implies that slope (m) = 6.

Step 3.

Obtain the value of y at x=3 by substituting x=3 into the original equation and the solving for y.

That is;

y = In(3² - 8)

= In( 9 - 8)

= In (1)

= 0

Hence, y=0 at x=3

Step 4

Obtain the y-intercept (c) by substituting m=6, y=0 and x=3 into y=mx + c

y=mx+c

0 = 6(3) + c

Step 5

Solve for c.

0= 18 + c

Subtract 18 from both-side of the equation.

-18 = c

Hence, c =-18

Step 6

Determine the equation by substituting m= 6 and c=-18 into y=mx + c

y = 6x - 18

Therefore, the equation of the tangent to the curve at the point where x = 3 is y = 6x - 18