Question

Use the limit definition of the derivative to find {d x}(5 x² -3 x)

Answer 1

To find the derivative of the function f(x) = 5x² - 3x using the limit definition, apply the definition and simplify to get f'(x) = 10x - 3, which is the derivative of the original function.

Step-by-step explanation:

You are interested in finding the derivative of the function f(x) = 5x² - 3x using the limit definition of the derivative. The limit definition of a derivative is f'(x) = lim_(h->0) (f(x+h) - f(x))/h. Let's apply this to the given function.

First, we apply the function to x + h:

• f(x + h) = 5(x + h)² - 3(x + h)

Now, we expand and simplify:

• f(x + h) = 5(x² + 2xh + h²) - 3x - 3h
• f(x + h) = 5x² + 10xh + 5h² - 3x - 3h

Next, we construct the difference quotient:

• (f(x + h) - f(x))/h = (5x² + 10xh + 5h² - 3x - 3h - (5x² - 3x))/h
• (f(x + h) - f(x))/h = (10xh + 5h² - 3h)/h
• (f(x + h) - f(x))/h = 10x + 5h - 3

Finally, we take the limit as h approaches 0:

• f'(x) = lim_(h->0) (10x + 5h - 3)
• f'(x) = 10x - 3

Therefore, the derivative of f(x) = 5x² - 3x is f'(x) = 10x - 3.