Solve the Derivatives
y = 3x^2
y' = 6x
∫ 2x dx
x^2 + C
lim (x -> 3) 2x - 1
5
y = x^2 at x = 1
y = 2x - 1
This rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g
Chain Rule
y = 2x^3 - 3x^2 + x + 5
y' = 6x^2 - 6x + 1
∫ (x^3 + 2x - 1) dx
(1/4)x^4 + x^2 - x + C
lim (x -> 0) (x^2 - 1) / (x - 1)
-1
y = 2x^2 + 3 at x = 2
y = 8x - 5
This rule is a formula used to find the derivatives of products of two or more functions.
Product Rule
y = e^(2x)
y' = 2e^(2x)
∫ (2x * e^x) dx
2x*e^x - 2e^x + C
lim (x -> 1) (sqrt(x) - 1) / (x - 1)
1/2
y = -x^2 + 4x + 3 at x = 3
y = -2x + 12
This rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number.
Power rule
y = sin(x) * cos(x)
y' = cos^2(x) - sin^2(x)
∫ e^(3x) * sin(2x) dx
1/5 * e^(3x) * (3sin(2x) - 2cos(2x)) + C
lim (x -> 0) sin(x) / x
1
y = 3x^2 - 2x + 1 at x = 1
y = 4x - 2
This rule says that the limit when we divide one function by another is the same after we take the derivative of each function
L’hopitals Rule
y = ln(x^2 + 1)
y' = 2x / (x^2 + 1)
∫ x^4 / (1 + x^2) dx
1/2 * x^3 - x + arctan(x) + C
lim (x -> 0) (e^x - 1 - x) / x^2
1/2
y = x^2 - 4x + 7 at x = 2
y = 3
Thi rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.
Quotient rule