Use the first principles definition to differentiate y = 5x^2. Please show all your work!
To differentiate the function y = 5x^2 using the first principles definition, we need to find the derivative using the limit definition of the derivative. Here's how we can do it step by step:
Step 1: Recall the first principles definition of differentiation:
The derivative of a function f(x) at a given point x = a is defined as:
f'(a) = lim(h -> 0)[f(a + h) - f(a)]/h
Step 2: Substitute the given function y = 5x^2 into the first principles definition:
y' = lim(h -> 0)[(5(a + h)^2 - 5a^2)/h]
Step 3: Expand the bracket and simplify:
y' = lim(h -> 0)[(5(a^2 + 2ah + h^2) - 5a^2)/h]
y' = lim(h -> 0)[(5a^2 + 10ah + 5h^2 - 5a^2)/h]
y' = lim(h -> 0)[(10ah + 5h^2)/h]
y' = lim(h -> 0)[10a + 5h]
Step 4: Evaluate the limit as h approaches 0:
Since h approaches 0, we can substitute h with 0 in the expression:
y' = 10a + 5(0)
y' = 10a
Step 5: Simplify the final expression:
The derivative of y = 5x^2 with respect to x is 10x.
Therefore, by applying the first principles definition of differentiation, we have found that the derivative of y = 5x^2 is 10x.