# Differentiate the following from first principles: (a) #y=10x# (b) #y=6-5x+x^2#?

a)

b)

By definition of the derivative:

Part (A)

Part (B)

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(a) The derivative of y = 10x from first principles is:

dy/dx = lim(h -> 0) [(10(x + h) - 10x) / h] = lim(h -> 0) [(10x + 10h - 10x) / h] = lim(h -> 0) [10h / h] = lim(h -> 0) 10 = 10

(b) The derivative of y = 6 - 5x + x^2 from first principles is:

dy/dx = lim(h -> 0) [(6 - 5(x + h) + (x + h)^2 - (6 - 5x + x^2)) / h] = lim(h -> 0) [(6 - 5x - 5h + x^2 + 2xh + h^2 - 6 + 5x - x^2) / h] = lim(h -> 0) [(5x^2 - 5h + 2xh + h^2) / h] = lim(h -> 0) [5x^2/h - 5 + 2x + h] = 2x - 5

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(a) To differentiate ( y = 10x ) from first principles:

Using the definition of the derivative from first principles:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute the function ( f(x) = 10x ):

[ f'(x) = \lim_{h \to 0} \frac{10(x + h) - 10x}{h} ]

Simplify:

[ f'(x) = \lim_{h \to 0} \frac{10x + 10h - 10x}{h} ] [ f'(x) = \lim_{h \to 0} \frac{10h}{h} ]

Cancel out ( h ):

[ f'(x) = \lim_{h \to 0} 10 ]

Evaluate the limit:

[ f'(x) = 10 ]

Therefore, the derivative of ( y = 10x ) with respect to ( x ) from first principles is ( 10 ).

(b) To differentiate ( y = 6 - 5x + x^2 ) from first principles:

Using the definition of the derivative from first principles:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute the function ( f(x) = 6 - 5x + x^2 ):

[ f'(x) = \lim_{h \to 0} \frac{6 - 5(x + h) + (x + h)^2 - (6 - 5x + x^2)}{h} ]

Expand and simplify:

[ f'(x) = \lim_{h \to 0} \frac{6 - 5x - 5h + x^2 + 2xh + h^2 - 6 + 5x - x^2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-5h + 2xh + h^2}{h} ]

Factor out ( h ):

[ f'(x) = \lim_{h \to 0} \frac{h(-5 + 2x + h)}{h} ]

Cancel out ( h ):

[ f'(x) = \lim_{h \to 0} (-5 + 2x + h) ]

Evaluate the limit:

[ f'(x) = -5 + 2x ]

Therefore, the derivative of ( y = 6 - 5x + x^2 ) with respect to ( x ) from first principles is ( -5 + 2x ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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