How do you use the limit definition of the derivative to find the derivative of #y=-2x+5#?

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Answer 1 · 2017-01-01T16:03:53

# dy/dx = -2 #

The definition of the derivative of #y=f(x)# is

# f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h #

So Let # f(x) = -2x+5 # then;

# f(x+h) = -2(x+h) + 4 #
# " "= -2x-2h + 4 #

And so #f(x+h)-f(x)# is given by:

# f(x+h)-f(x) = (-2x-2h + 4 ) - (-2x+4) #
# " "= -2x-2h + 4 +2x-4 #
# " "= -2h #

And so the derivative of #y=f(x)# is given by:

# \ \ \ \ \ dy/dx = lim_(h rarr 0) (-2h) / h #
# " " = lim_(h rarr 0) -2 #
# " " = -2 #
# :. dy/dx = -2 #