Another question,
if f(x+h)-f(x)= -3hx**2+4hx-8h**2x+5h**2-3h**3,
what is f'(x)?
Looks like you are learning derivatives from First Principles.
By definition,
f'(x) = limit [(f(x+h)-f(x))/h] as h --> 0
you were given the f(x+h)-f(x) part as
-3hx^2 + 4hx - (8h^2)x + 5h^2 - 3h^3 , (I read your ** as exponents)
so f'(x)
= limit(-3hx^2 + 4hx - (8h^2)x + 5h^2 - 3h^3)/h as h---0
= limit (-3x^2 + 4x - (8h)x + 5h - 3h^2 as h ---> 0
= -3x^2 + 4x
To find f'(x), we need to differentiate the given expression with respect to x.
Let's differentiate each term step by step:
1. Differentiate -3hx^2 with respect to x:
The power rule for differentiation states that the derivative of x^n, where n is a constant, is nx^(n-1).
Applying this rule, the derivative of -3hx^2 with respect to x is -3h * 2x^(2-1) = -6hx.
2. Differentiate 4hx with respect to x:
The derivative of 4hx with respect to x is 4h.
3. Differentiate -8h^2x^2 with respect to x:
The constant -8h^2 is not affected by differentiation, so the derivative of -8h^2x^2 with respect to x is -8h^2 * 2x^(2-1) = -16h^2x.
4. Differentiate 5h^2 with respect to x:
The constant 5h^2 is not affected by differentiation, so the derivative is 0.
5. Differentiate -3h^3 with respect to x:
The constant -3h^3 is not affected by differentiation, so the derivative is 0.
Now, let's add up the derivatives of each term we found:
f'(x) = -6hx + 4h - 16h^2x + 0 + 0
Simplifying the expression, we get:
f'(x) = -16h^2x - 6hx + 4h
To find the derivative of the function f(x), denoted as f'(x), we need to use the definition of the derivative. The derivative of f(x) gives us the rate of change of the function at any given point.
In this case, we have the expression f(x+h) - f(x) on the left side. This is similar to the definition of the difference quotient, which is used to find the derivative.
The difference quotient is given by:
[f(x+h) - f(x)] / h
Comparing this with the given expression, we can see that the function f(x) is expressed as:
f(x) = -3hx^2 + 4hx - 8h^2x + 5h^2 - 3h^3
Now, we can substitute this expression into the difference quotient:
[f(x+h) - f(x)] / h = [-3h(x+h)^2 + 4h(x+h) - 8h^2(x+h) + 5h^2 - 3h^3 - (-3hx^2 + 4hx - 8h^2x + 5h^2 - 3h^3)] / h
Simplifying this expression, we get:
[f(x+h) - f(x)] / h = [-3hx^2 - 6hxh - 3hh^2 + 4hx + 4hh - 8h^2x - 8h^2 + 5h^2 - 3h^3 + 3hx^2 - 4hx + 8h^2x - 5h^2 + 3h^3] / h
The common terms cancel out, leaving us with:
[f(x+h) - f(x)] / h = (4hh - 3hh^2 + 4hx - 4hx + 8h^2x - 8h^2x + 5h^2 - 5h^2 + 3h^3 - 3h^3) / h
Now, we can simplify further:
[f(x+h) - f(x)] / h = (4hh - 3hh^2 + 3h^3) / h
Dividing each term by h, we get:
[f(x+h) - f(x)] / h = 4h - 3h^2 + 3h^2
Simplifying this expression gives us:
[f(x+h) - f(x)] / h = 4h
Now, taking the limit as h approaches 0, we can find the derivative:
lim(h -> 0) [f(x+h) - f(x)] / h = lim(h -> 0) 4h
The limit of 4h as h approaches 0 is simply 0.
Therefore, we can conclude that f'(x) = 0.