Derivatives
Find the derivative of f(x) = 4x^5 − 5x^4
f ′ (x) = 20x^4 − 20x^3
Integrate F'(x)=3x^2+6x dx
x^3+3x^2+C
When is a function differentiable?
A function is considered differentiable if its derivative exists at each point in its domain
Find the derivative of f(x) = e^x * sin (x)
f ′ (x) = e^x * cos(x) + e^x * sin(x)
Integrate F'(x)= (4x^2-8x+1) dx
4x^3 / 3 - 4x^2 + x + C
What is a midpoint Riemann sum?
The average of the left and right endpoints where the point is then evaluated to find the height of each rectangle.
Find the derivative of f(x)= √ (x^2+ 8)
f ′ (x) = x / √(x^2 + 8)
Integrate F'(x) = (3x^5 - x^5/3) dx
x^6 /2 - 3x^8/3 / 8 + C
What does the Mean Value Theorem state?
If a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]
Find the derivative of f(x)=tan(3x)
f ′ (x) = 3sec2(3x)
Integrate F'(x) = (x^3 + 3x^2 -9x -2) / (x-2) dx
x^3 / 3 + 5x^2/2 + x + C
What is the chain rule?
The derivative of the outside with the inside plugged in times the derivative of the inside
Find the derivative of f(x)=6/ (3x^2 − π(pi))^4
f ′ (x) = -144x / (3x^2 - π)^5
Integrate F'(x) = x / (4+x^2)^2 dx
-1 / (8 + 2x^2) + C
What is the difference between IVT and MVT?
The mean value theorem guarantees that the derivatives have certain values, whereas the intermediate value theorem guarantees that the function has certain values between two given values.