Derivative of Exponential Functions base e

Front

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

Back

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

Front

What is the general rule for differentiating exponential functions of the form f(x) = ae^(bx)?

Back

The derivative is f'(x) = abe^(bx), where a and b are constants.

Front

How do you find the derivative of a product of functions, such as f(x) = xe^x?

Back

Use the product rule: f'(x) = u'v + uv', where u = x and v = e^x.

Front

What is the derivative of the function f(x) = 5e^(3x)?

Back

The derivative is f'(x) = 15e^(3x).

Front

What is the second derivative of the function f(x) = 2x - 5x^6?

Back

The second derivative is f''(x) = 2 - 30x^5.

Front

If y = e^(-2x^3), how do you find dy/dx?

Back

Use the chain rule: dy/dx = -6x^2e^(-2x^3).

Front

What is the chain rule in differentiation?

Back

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

Front

What is the derivative of the function f(x) = 10e^x?

Back

The derivative is f'(x) = 10e^x.

Front

What is the significance of the base 'e' in exponential functions?

Back

The base 'e' is an irrational number approximately equal to 2.71828, and it is the unique base for which the derivative of e^x is equal to e^x.

Front

How do you differentiate an exponential function with a negative exponent, such as f(x) = e^(-x)?

Back

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