Derivatives of exponential functions

Front

What is the derivative of the function y = e^(kx)?

Back

The derivative is f'(x) = ke^(kx), where k is a constant.

Front

How do you find the derivative of a natural logarithm function, y = ln(u)?

Back

Use the chain rule: f'(x) = (1/u) * (du/dx), where u is a function of x.

Front

What is the derivative of y = 3ln(x^2 - 3)?

Back

The derivative is f'(x) = 6x/(x^2 - 3).

Front

What is the derivative of the function f(x) = 2e^(-3x)?

Back

The derivative is f'(x) = -6e^(-3x).

Front

What is the derivative of y = e^(4x)?

Back

The derivative is f'(x) = 4e^(4x).

Front

How do you differentiate a product of functions, such as y = ln(3x^2 - 5x)?

Back

Use the quotient rule: f'(x) = (6x - 5)/(3x^2 - 5).

Front

What is the general rule for differentiating exponential functions of the form y = e^(kx)?

Back

The derivative is f'(x) = ke^(kx), where k is a constant.

Front

What is the derivative of y = e^(3x)?

Back

The derivative is f'(x) = 3e^(3x).

Front

What is the chain rule in differentiation?

Back

The chain rule states that if you have a composite function y = f(g(x)), then the derivative is f'(g(x)) * g'(x).

Front

What is the product rule in differentiation?

Back

The product rule states that if you have two functions u(x) and v(x), then the derivative is u'v + uv'.