Using the definition of derivative , find the derivative of f(x)=9-3x

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative of f(x) is defined as lim h-->0 [ f (x+h) - f(x)]/h

Here f(x) = 9 - 3x

Substituting this in lim h-->0 [ f (x+h) - f(x)]/h

=> lim h-->0 [ 9-3(x+h) - 9 + 3x]/h

=> lim h-->0 [ 9-3x -3h - 9 + 3x]/h

=> lim h-->0 [-3h]/h

=> lim h-->0 (-3)

There is no h left to substitute, therefore f'(x) = -3

The required derivative is -3.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The definition of derivative states that:

f'(x) = lim [f(x+h) - f(x)]/h, for h->0

We'll calculate f(x+h) = 9 - 3(x+h)

We'll remove the brackets using distributive property:

f(x+h) = 9 - 3x - 3h

We'll substitute the expressions of f(x+h) and f(x) in the definition of derivative:

f'(x) = lim (9 - 3x - 3h - 9 + 3x)/h , for h->0

We'll combine and eliminate like terms:

f'(x) = lim -3h/h

We'll simplify and we'll get:

f'(x) = lim -3

f'(x) = -3

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