Derivatives of Logs and Exponentials

Front

What is the derivative of the function \( e^{u} \) with respect to \( x \), where \( u \) is a function of \( x \)?

Back

The derivative is \( e^{u} \cdot \frac{du}{dx} \).

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 \( f'(g(x)) imes g'(x) \).

Front

What is the derivative of \( \ln(x) \)?

Back

The derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Front

How do you differentiate \( \ln(a(x)) \) where \( a(x) \) is a function of \( x \)?

Back

The derivative is \( \frac{a'(x)}{a(x)} \).

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 \( u'v + uv' \).

Front

What is the quotient rule in differentiation?

Back

The quotient rule states that if you have a function \( \frac{u(x)}{v(x)} \), then the derivative is \( \frac{u'v - uv'}{v^2} \).

Front

What is the derivative of \( e^{kx} \) where \( k \) is a constant?

Back

The derivative is \( ke^{kx} \).

Front

What is the derivative of \( \sin(u) \) where \( u \) is a function of \( x \)?

Back

The derivative is \( \cos(u) \cdot \frac{du}{dx} \).

Front

What is the derivative of \( \cos(u) \) where \( u \) is a function of \( x \)?

Back

The derivative is \( -\sin(u) \cdot \frac{du}{dx} \).

Front

How do you differentiate \( \ln\left(\frac{f(x)}{g(x)}\right) \)?

Back

The derivative is \( \frac{f'(x)g(x) - f(x)g'(x)}{f(x)g(x)} \).