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: $f'(x) = -3x^{-4} = \frac{-3}{x^4}$

$f(x) = x^n$, then $f'(x) = nx^{n-1}$

If $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$

If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$

$\frac{d}{dx} \sin(x) = \cos(x)$

$\frac{d}{dx} \cos(x) = -\sin(x)$

$\frac{d}{dx} e^x = e^x$

$\frac{d}{dx} \ln(x) = \frac{1}{x}$

$\frac{d}{dx} a^x = a^x \ln(a)$

$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(y)}$

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.