The instantaneous rate of change of a function.

A method to find the derivative of a function where y is not explicitly defined in terms of x.

A rule for differentiating composite functions.

A rule to differentiate functions of the form $x^n$.

A rule for differentiating the product of two functions.

A rule for differentiating the quotient of two functions.

A function that reverses another function.

A function formed by substituting one function into another.

A line that touches a curve at a single point.

The derivative of a constant times a function is the constant times the derivative of the function.

The chain rule is used to find the derivative of a composite function. It states that the derivative of $f(g(x))$ is $f'(g(x)) * g'(x)$.

The product rule is used to find the derivative of two functions multiplied together. It states that the derivative of $f(x)g(x)$ is $f'(x)g(x) + f(x)g'(x)$.

The quotient rule is used to find the derivative of two functions divided by each other. It states that the derivative of $\frac{f(x)}{g(x)}$ is $\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.

Implicit differentiation is used when it is difficult or impossible to solve for y explicitly in terms of x. You differentiate both sides of the equation with respect to x, remembering to use the chain rule when differentiating terms involving y, and then solve for $\frac{dy}{dx}$.

The derivative of an inverse function $f^{-1}(x)$ is given by $\frac{1}{f'(f^{-1}(x))}$, where $f'(x)$ is the derivative of the original function.

The power rule states that if $f(x) = x^n$, then the derivative $f'(x) = nx^{n-1}$.

The constant multiple rule states that if $f(x) = cf(x)$, then $f'(x) = cf'(x)$

The sum/difference rule states that if $h(x) = f(x) \pm g(x)$, then $h'(x) = f'(x) \pm g'(x)$

If $f$ and $g$ are inverse functions, then $f(g(x)) = x$ and $g(f(x)) = x$.

The derivative of a constant is always zero.

Apply the chain rule: $f'(x) = \cos(x^2) \cdot 2x$

Apply the product rule: $f'(x) = 3x^2e^x + x^3e^x$

Apply the quotient rule: $f'(x) = \frac{2x\cos(x) - x^2(-\sin(x))}{\cos^2(x)}$

Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$. Solve for $\frac{dy}{dx}: \frac{dy}{dx} = -\frac{x}{y}$

Use the chain rule: $f'(x) = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$

Apply the chain rule: $f'(x) = 5(x^2 + 3x)^4 \cdot (2x + 3)$

Apply the chain rule: $f'(x) = \sec^2(3x) \cdot 3$

Apply the chain rule and constant multiple rule: $f'(x) = 5e^{2x} \cdot 2 = 10e^{2x}$

Rewrite as $f(x) = (4x+1)^{1/2}$ then apply the chain rule: $f'(x) = \frac{1}{2}(4x+1)^{-1/2} \cdot 4 = \frac{2}{\sqrt{4x+1}}$

Rewrite as $f(x) = x^{-3}$ then apply the power rule: $f'(x) = -3x^{-4} = \frac{-3}{x^4}$