Rates of Change - AP Precalculus

$f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$. Negative of arcsecant derivative.

$f'(x) = -\text{csc}(x)\text{cot}(x)$. Quotient rule applied to $\csc(x) = \frac{1}{\sin(x)}$.

$f'(x) = -\text{csc}^2(x)$. Cotangent is $\frac{\cos(x)}{\sin(x)}$, use quotient rule.

Rate of change is 5. Use $f'(x) = 2x + 3$, evaluate at $x = 1$.

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

$f'(x) = 5^x \ln(5)$. General formula $a^x$ derivative includes $\ln(a)$ factor.

$f'(x) = 2x$. Apply power rule: bring down exponent 2, reduce power by 1.

$\frac{f(b) - f(a)}{b - a}$. Slope formula between two points on the function.

The derivative at a specific point. The slope of tangent line at one point.

$f'(x) = -\text{sin}(x)$. Cosine differentiates to negative sine.

$f'(x) = 2\text{e}^{2x}$. Chain rule: multiply by inner derivative 2.

Rate of change is 3. Use $f'(x) = 3x^2$, evaluate at $x = 1$.

$f'(x) = e^x$. Exponential function $e^x$ is its own derivative.

Rate of change is 0. Constant functions have zero rate of change.

$f'(x) = \text{sec}^2(x)$. Tangent differentiates to secant squared.

Derivative of a constant is 0. Constants have zero slope everywhere.

$10x$. Factor out constant 5, apply power rule.

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

$f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent $\frac{1}{2}$.

Rate of change is 32. Use $f'(x) = 4x^3$, evaluate at $x = 2$.

$f'(x) = -\text{csc}(x)\text{cot}(x)$. Quotient rule applied to $\csc(x) = \frac{1}{\sin(x)}$.

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

$5x^4 - 6x^2$. Apply power rule to each term separately.

$5x^4 - 6x^2$. Apply power rule to each term separately.

$f'(x) = \frac{1}{\sqrt{1-x^2)}$. Standard inverse trig derivative formula.

$f'(x) = \text{cos}(x)$. Sine differentiates to cosine.

$f'(x) = x^x(\text{ln}(x)+1)$. Use logarithmic differentiation for variable base and exponent.

$f'(x) = 2\text{sin}(x)\text{cos}(x)$. Use chain rule on $\sin^2(x)$.

$f'(x) = -2\text{sin}(x)\text{cos}(x)$. Chain rule gives negative of $\sin^2(x)$ derivative.