Given function fx 3x2 5x 29 Find fx h fx hSolutionGi

Given function ? = f(x) = - 3x^2 - 5x + 29. Find f(x+ h) - f(x) / h

Given,

f(x) = -3x^2 - 5x +29

So, f(x+h) = -3(x+h)^2 - 5(x+h) +29 Plug (x+h) in place of x

=> {f(x+h)-f(x)} / h = {-3(x+h)^2 - 5(x+h) +29 - (-3x^2 - 5x +29)} / h

Apply formula (a+b)^2 = a^2 + 2ab + b^2

=> {f(x+h)-f(x)} / h = {-3(x^2 + 2xh + h^2) - 5x - 5h + 29 + 3x^2 + 5x - 29} / h

=> (f(x+h)-f(x)} / h = { -3x^2 - 6xh -3h^2 - 5x - 5h + 29 + 3x^2 + 5x - 29} / h

Simplify like terms

=> (f(x+h)-f(x)} / h = { -6xh - 3h^2 - 5h } / h

take -h common from -6xh - 3h^2 - 5h

=> (f(x+h)-f(x)} / h = -h(6x + 3h +5) / h

h in the numerator will get cancelled with h in the denominator

=> (f(x+h)-f(x)} / h = -(6x + 3h +5)