Using the limit definition, how do you differentiate #f(x)=x^2+3x+1#?
In this case
By signing up, you agree to our Terms of Service and Privacy Policy
To differentiate the function f(x) = x^2 + 3x + 1 using the limit definition of the derivative, follow these steps:
-
Write down the definition of the derivative using the limit: f'(x) = lim(h->0) [f(x + h) - f(x)] / h
-
Substitute the function f(x) = x^2 + 3x + 1 into the formula: f'(x) = lim(h->0) [(x + h)^2 + 3(x + h) + 1 - (x^2 + 3x + 1)] / h
-
Expand and simplify the expression: f'(x) = lim(h->0) [(x^2 + 2hx + h^2) + 3x + 3h + 1 - x^2 - 3x - 1] / h = lim(h->0) [2hx + h^2 + 3h] / h
-
Cancel out common terms and simplify further: f'(x) = lim(h->0) [2x + h + 3] = 2x + 3
Therefore, the derivative of f(x) = x^2 + 3x + 1 using the limit definition is f'(x) = 2x + 3.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- For #f(x)=x^2/sqrt(x^3-5)# what is the equation of the tangent line at #x=5#?
- What is the equation of the tangent line of #y=1/x^2# at #x=1#?
- How do you estimate the instantaneous rate of change at the point for #x=5# for #f(x) = ln(x)#?
- What is the equation of the tangent line for #sin(x)# at the point x=2?
- How do you find the derivative of #-5x^2+8x+2# using the limit definition?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7