Use the four step process to find the slope of the tangent line to the graph of the given function at any point. ( simplify your answer is completely)
f(x)= 8x^2 +3x
i’ve been trying to do this problem for 20 minutes,

Here, we are required to find the slope of the tangent line to the graph of the given function at any point.

The correct answer is 16x + 3.

The four step process to find the slope of the tangent line to the graph at any point is as follows;

First step (1):

First, there's a need to evaluate the value of F(x+h), the result of which is;

f(x +h) = 8(x + h)² + 3( x + h) and yields;

f(x + h) = 8( x² + 2xh + h²) + 3( x + h)

and, f (x + h) = 8x² + 16xh + 8h² + 3x + 3h.

Second step(2):

Second step involves subtraction of F(x) from F(x+h), i.e

F(x+h) - f(x) = 8x² + 16xh + 8h² + 3x + 3h - ( 8x² + 3x)

and, F(x+h) - f(x) = 16xh + 8h² + 3 h

Third step (3):

The third step involves dividing both sides of the equation by h, i.e

{F(x+h) - f(x)} / h = (16xh)/h + (8h²)/h + (3 h)/h

This in turn yields;

{F(x+h) - f(x)} / h = 16x + 8h + 3.

Fourth step(4):

This step involves taking limits on both sides as h => 0., i.e

lim f(x + h) - f(x)/ h =

h=>0

lim 16x + 8h + 3 = lim 16x + 8(0) + 3 = 16x + 3

h=>0. h=>0

Therefore, the slope of the tangent line to the graph of the given function at any point is;

16x + 3.

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The answer to your question is: 16x + 3

Step-by-step explanation:

Step 1 :      f(x) = 8x² + 3x

                f(x +h) =  8(x + h)² + 3( x + h)

                f(x + h) = 8( x² + 2xh + h²) + 3( x + h)

                f (x + h) = 8x² + 16xh + 8h² + 3x + 3h

Step 2 f(x+h) - f(x) =  8x² + 16xh + 8h² + 3x + 3h - ( 8x² + 3x)

                              = 8x² + 16xh + 8h² + 3x + 3h - 8x² -3x

                              = 16xh + 8h² + 3 h

Step 3 f(x + h) - f(x)/ h = h(16x + 8h + 3) /h

                                    = 16x + 8h + 3

Step 4 lim  f(x + h) - f(x)/ h = lim 16x + 8h + 3  = lim 16x + 8(0) + 3 = 16x + 3

        h ⇒0                           h ⇒0                     h ⇒0

1 Remove parentheses.

-1/2-5/9

2 Find the Least Common Denominator (LCD) of 1/2, 5/9 In other words, find the Least Common Multiple (LCM) of  2,9

LCD = 18

3 Make the denominators the same as the LCD.

-1 x 9 /2 x 9 +5 x 2 /9 x 2

4 Simplify. Denominators are now the same.

-9/18 +10/18

5 Join the denominators.

-9 + 10/ 18

6 Simplify.

1/18

and your answer is 1/18

Step-by-step explanation: