Selecting Procedures for Calculating Derivatives - AP Calculus BC

$(uv)' = u'v + uv'$. Product rule: first times derivative of second plus second times derivative of first.

$ (u/v)' = \frac{u'v - uv'}{v^2} $. Quotient rule formula for division of functions.

$f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x)\cos(x)$.

$\frac{d}{dx}[\text{cos}(x)] = -\text{sin}(x)$. Derivative of cosine is negative sine.

$f'(x) = 2x\text{sec}^2(x^2)$. Chain rule with tangent: $\sec^2(x^2) \cdot 2x$.

$\frac{d}{dx}[\text{tan}(x)] = \text{sec}^2(x)$. Derivative of tangent is secant squared.

If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by exponent, reduce power by 1.

$\frac{d}{dx}[\text{csc}(x)] = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative involves cotangent.

$\frac{d}{dx}[\text{cot}(x)] = -\text{csc}^2(x)$. Cotangent derivative is negative cosecant squared.

$f'(x) = 2\text{sec}^2(x)\text{sec}(x)\text{tan}(x)$. Chain rule: $2\sec(x) \cdot \sec(x)\tan(x)$.

$f'(x) = \frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{x^2+1} \cdot 2x$.

$f'(x) = -3\text{cos}^2(x)\text{sin}(x)$. Chain rule: $3\cos^2(x) \cdot (-\sin(x))$.

$f'(x) = \text{e}^{\text{sin}(x)}\text{cos}(x)$. Chain rule: $e^{\sin(x)} \cdot \cos(x)$.

$f'(x) = \frac{\text{e}^x(x-1)}{x^2}$. Quotient rule with $u = e^x, v = x$.

$f'(x) = \text{cos}(x) - \text{sin}(x)$. Sum rule: derivative of sum equals sum of derivatives.

$f'(x) = -3x^2\text{csc}(x^3)\text{cot}(x^3)$. Chain rule with cosecant function.

$f'(x) = -\frac{2x}{(x^2 + 1)^2}$. Quotient rule with $u = 1, v = x^2 + 1$.

$f'(x) = 2x\text{e}^{x^2}$. Chain rule: $e^{x^2} \cdot 2x$.

$f'(x) = 2x\text{sin}(x) + x^2\text{cos}(x)$. Product rule: $u'v + uv'$ where $u = x^2, v = \sin(x)$.

$f'(x) = 2e^{2x}$. Chain rule: derivative of inside times $e^{2x}$.

$f'(x) = 12x^3 + 4x$. Apply power rule to each term separately.

$\frac{d}{dx}[\text{sec}(x)] = \text{sec}(x)\text{tan}(x)$. Secant derivative involves tangent.

$f'(x) = 4x^3 - 6x^{-3}$. Power rule applied to positive and negative exponents.

$f'(x) = 2\text{sec}(2x)\text{tan}(2x)$. Chain rule with secant function.

$f'(x) = \frac{1}{x}$. Chain rule: $\frac{d}{dx}[\ln(2x)] = \frac{1}{2x} \cdot 2 = \frac{1}{x}$.

$f'(x) = \frac{5}{3}x^{2/3}$. Power rule with fractional exponent.

$\frac{d}{dx}[\text{ln}(x)] = \frac{1}{x}$. Natural log derivative is reciprocal function.

$\frac{d}{dx}[\text{sin}(x)] = \text{cos}(x)$. Derivative of sine is cosine.

$f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.