2024 Derivative using Chain Rule& Higher Order (over 2 days)

Front

What is the Chain Rule in calculus?

Back

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have two functions f(x) and g(x), then the derivative of their composition f(g(x)) is f'(g(x)) * g'(x).

Front

How do you find the derivative of a function raised to a power, such as f(x) = (g(x))^n?

Back

To find the derivative of a function raised to a power, use the Chain Rule: f'(x) = n * (g(x))^(n-1) * g'(x).

Front

What is the derivative of f(x) = (x^6 + 4)^5?

Back

f'(x) = 30x^5(x^6 + 4)^4, using the Chain Rule.

Front

How do you differentiate trigonometric functions like y = csc(u)?

Back

The derivative of y = csc(u) is y' = -csc(u)cot(u) * du/dx, where u is a function of x.

Front

What is the derivative of y = csc(8x^2)?

Back

dy/dx = -16xcsc(8x^2)cot(8x^2), applying the Chain Rule.

Front

What is the formula for the derivative of a polynomial function?

Back

The derivative of a polynomial function f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0 is f'(x) = n*a_n*x^(n-1) + (n-1)*a_(n-1)*x^(n-2) + ... + a_1.

Front

How do you find the derivative of a function like y = (2x + 1)^10?

Back

Use the Chain Rule: y' = 10(2x + 1)^9 * 2 = 20(2x + 1)^9.

Front

What is the derivative of a constant?

Back

The derivative of a constant is zero.

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Front

How do you differentiate a composite function?

Back

To differentiate a composite function, apply the Chain Rule: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).