Find the derivative of f(x) = −2x+3 from first principles.

Question

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To determine f'(x) from first principles, we use the definition of the derivative: f'(x) = lim_(h → 0) (f(x+h) − f(x))/(h); Given f(x) = −2x+3. Step 1: Find f(x+h). Substitute x+h into the function f(x): f(x+h) = −2(x+h) + 3; f(x+h) = −2x − 2h + 3 Step 2: Find the difference f(x+h) − f(x). f(x+h) − f(x) = (−2x − 2h + 3) − (−2x + 3); f(x+h) − f(x) = −2x − 2h + 3 + 2x − 3; f(x+h) − f(x) = −2h Step 3: Form the difference quotient (f(x+h) − f(x))/(h). (f(x+h) − f(x))/(h) = (−2h)/(h); (f(x+h) − f(x))/(h) = −2 Step 4: Take the limit as h → 0. f'(x) = lim_(h → 0) (−2) Since the expression does not contain h, the limit is the constant value itself. f'(x) = −2 The derivative from first principles is −2.

Point	x	y
y-intercept	0	3
x-intercept	1.5	0

Find the derivative of f(x) = -2x+3 from first principles.

Answer

Step-by-step solution

Find $f(x+h)$ .

Find the difference $f(x+h) - f(x)$ .

Form the difference quotient $\frac{f(x+h) - f(x)}{h}$ .

Take the limit as $h \to 0$ .

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