Math Question?

j(x) = 10x^2 - 1/4x

(j(-4 + h) - j(-4))/h

= [10(h - 4)^2 - 1/4(h - 4) - 160 + 1/16)]/h

= 10 h + 17/(16 h) - 321/4

Ambiguity: 10x^2-x/4 or 10x^2-1/(4x)?

Make clear it & re-post your question.

Taking that as-written:

j(x) = 10x² - (1/4)x

You want to simplify:

[j(-4 + h) - j(-4)] / h

Let's start with:

j(x + h), first:

j(x) = 10x² - (1/4)x

j(x + h) = 10(x + h)² - (1/4)(x + h)

j(x + h) = 10(x² + 2hx + h²) - (1/4)x - (1/4)h

j(x + h) = 10x² + 20hx + 10h² - (1/4)x - (1/4)h

Now we can substitute these into:

[j(x + h) - j(x)] / h

[10x² + 20hx + 10h² - (1/4)x - (1/4)h - (10x² - (1/4)x)] / h

Simplifying that:

[10x² + 20hx + 10h² - (1/4)x - (1/4)h - 10x² + (1/4)x] / h

And the x² and x terms cancel out:

[20hx + 10h² - (1/4)h] / h

Now we can factor out and cancel the h:

h (20x + 10h - 1/4) / h

20x + 10h - 1/4

Now set x = -4 and simplify:

20(-4) + 10h - 1/4

-80 + 10h - 1/4

-320/4 + 10h - 1/4

10h - 321/4

So I do get something different than you.

j(-4+h) = 10(-4 + h)^2 - (1/4)(-4 + h)

. . . . . . = 10(16 - 8h + h^2) + (4/4) - h/4

. . . . . . = 160 - (80h + h/4) + 10h^2  + 1

. . . . . . = 161 - (321/4)h + 10h^2

j(-4) . = 10(-4^2) - (1/4)(-4)

. . . . = 160 - (-4/4) = 161

Hence j(-4+h) - j(-4) = (161 - (321/4)h + 10h^2) - 161

. . . . . . . . . . . . . . . . . = 10h^2 - (321/4)h

. . . . . . . . . . j(-4+h) - j(-4)

and . . . . . . ------------------ . =�� . 10h - (321/4)

. . . . . . . . . . . . . . h

f(x) = 10x² - 1/4x

f(x+h) = 10(x+h)² - 1/4(x+h)

D(x) = Δf(x)/Δx = [f(x+h) - f(x)]/h

 = [10(x+h)² - 1/4(x+h) - 10x² + 1/4x]/h

 = [10(x² + 2xh + h²) - x/4 - h/4 - 10x² + x/4]/h

 = [10x² - 10x² + 20xh + 10h² - x/4 + x/4 - h/4]/h

 = [ 20xh + 10h² - h/4]/h

D(x) = 20x + 10h - 1/4

D(-4) = -80 - 1/4 + 10h

 = 10h - 321/4

guessing that is

j(x) = 10x² – (1/4)x

not 10x² – 1/(4x)

j(-4+h) = 10(-4+h)² – (1/4)(-4+h)

j(-4+h) = 10(16+h²–8h) + 1 – h/4

j(-4+h) = 160 + 10h² – 80h + 1 – h/4

j(-4+h) = 161 + 10h² – 80h – h/4

j(-4) = 10(-4)² – (1/4)(-4)

j(-4) = 160 + 1 = 161

next difficulty, you list

j(-4+h) – j(-4)/h

which is

j(-4+h) – (j(-4)/h)

which is

161 + 10h² – 80h – h/4 – 161/h

but I suspect you mean (incorrectly)

(j(-4+h) – j(-4)) / h

j(-4+h) – j(-4) = 161 + 10h² – 80h – h/4 – 161j(-4+h) – j(-4) = 10h² – 80h – h/4

(j(-4+h) – j(-4)) / h = 10h – 80 – 1/4 = 10h – 321/4

please use parans correctly.