Derivative Notation - AP Calculus BC

$\frac{1}{x}$. Natural log derivative is reciprocal function.

$-\sin x$. Derivative of cosine is negative sine.

$f'(x) = 6x + 2$. Apply power rule to each term separately.

$ - \csc^2 x $. Derivative of cotangent is negative cosecant squared.

Slope of the tangent line. Derivative gives instantaneous rate of change.

$f'(x) = \frac{3}{2}x^{1/2}$. Rewrite $\sqrt{x^3} = x^{3/2}$ and use power rule.

$f'(x) = -x^{-3}$. Constant $\frac{1}{2}$ times power rule on $x^{-2}$.

$f'(x) = \cos x - x \sin x$. Product rule with $u = x$ and $v = \cos x$.

$f'(x) = 5e^x$. Constant multiple rule with exponential derivative.

$f'(x) = -4x^{-3/2}$. Constant 8 times power rule on $x^{-1/2}$.

$f'(x) = x^2$. Constant factor $\frac{1}{3}$ remains, apply power rule.

$f'(x) = 2\sec x \tan x$. Constant multiple of secant derivative.

$f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and use power rule.

$ - \csc x \cot x$. Derivative is negative product of cosecant and cotangent.

$f'(x) = \frac{x^2 - 1}{x^2}$. Rewrite as $x + x^{-1}$ and differentiate.

$\frac{dy}{dx}$. Leibniz notation shows derivative of $y$ with respect to $x$.

$f'(x) = 4x^3 - 6x + 1$. Apply power rule to each term.

$\frac{2x}{x^2 + 1}$. Use chain rule with $\ln$ and $x^2 + 1$.

$f'(x) = \frac{1}{2\sqrt{x}}$. Rewrite as $x^{1/2}$ and apply power rule.

$\sec^2 x$. Derivative of tangent is secant squared.

$f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}$. Limit of difference quotient as $h$ approaches 0.

$a^x \ln a$. Exponential with base $a$ requires $\ln a$ factor.

$f'(x) = 3x^2 - 5$. Derivative of constant is 0, apply power rule to other terms.

$f'(x) = \cot x$. Chain rule with $\ln(\sin x)$.

$f'(x) = 2x \sec^2(x^2)$. Chain rule: outer derivative times inner derivative.

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

$f'(x) = 2\sin x \cos x$. Use chain rule with $\sin^2 x = (\sin x)^2$.

$f'(x) = 35x^4$. Constant multiple of power rule.

$2x \sin x + x^2 \cos x$. Apply product rule: $(uv)' = u'v + uv'$.