Tag Archives: Differentiation

How to Express the Velocity and the Acceleration as Functions of Displacement and Time

How to Express the Velocity and the Acceleration as Functions of Displacement and Time

If a particle the displacement gives $P$ moves in a straight line and its position function $x(t)$, then: $$x(t) \xrightarrow{\text{differentiate}}v(t)=\dfrac{dx}{dt}\xrightarrow{\text{differentiate}} a(t)=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$$ Sign Interpretation Suppose a particle $P$ moves in a straight line with displacement function $s(t)$ relative to an origin $O$. Its velocity function is $v(t)$, and its acceleration function is $a(t)$.The sign diagram is […]

Turning Points and Nature

Turning Points and Nature

A function’s turning point is where $f'(x)=0$. A maximum turning point is a turning point where the curve is concave up (from increasing to decreasing ) and $f^{\prime}(x)=0$ at the point.$$ \begin{array}{|c|c|c|} \hlinef^{\prime}(x) \gt 0 & f'(x) = 0 & f'(x) \lt 0 \\ \hline& \text{maximum} & \\\nearrow & & \searrow \\ \hline\end{array} $$A minimum […]

Increasing Functions and Decreasing Functions

Increasing Functions and Decreasing Functions

Increasing and Decreasing We can determine intervals where a curve increases or decreases by considering $f'(x)$ on the interval in question. Monotone (Monotonic) Increasing or Decreasing Many functions increase or decrease for all $x \in \mathbb{R}$. These functions are called either monotone (monotonic) increasing or monotone (monotonic) decreasing. $y=2^x$ is monotone (monotonic) increasing for all […]

Higher Derivatives

Higher Derivatives

Given a function $f(x)$, the derivative $f^{\prime}(x)$ is known as the first derivative.The second derivative of $f(x)$ is the derivative of $f^{\prime}(x)$, which is $f^{\prime \prime}(x)$ or the derivative of the first derivative.$$ \displaystyle \begin{align}f^{\prime}(x) &= \dfrac{d}{dx}f(x) \\f^{\prime \prime}(x) &= \dfrac{d}{dx}f'(x) \\f^{(3)}(x) &= \dfrac{d}{dx}f^{\prime \prime}(x) \\f^{(4)}(x) &= \dfrac{d}{dx}f^{(3)}(x) \\&\cdots \\f^{(n)}(x) &= \dfrac{d}{dx}f^{(n-1)}(x) \\\end{align} $$We can […]

Derivative of Trigonometric Functions

Derivative of Trigonometric Functions

$$ \large \displaystyle \begin{align}\dfrac{d}{dx}\sin{x} &= \cos{x} \\\dfrac{d}{dx}\cos{x} &= -\sin{x} \\\dfrac{d}{dx}\tan{x} &= \sec^2{x} \\\end{align} $$ Example 1 Prove $\dfrac{d}{dx}\tan{x} = \sec^2{x}$ using $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\cos{x} = -\sin{x}$. \( \begin{align} \displaystyle \require{AMSsymbols} \require{AMSsymbols} \require{color}\dfrac{d}{dx}\tan{x} &= \dfrac{d}{dx}\dfrac{\sin{x}}{\cos{x}} \\&= \dfrac{\dfrac{d}{dx}\sin{x} \times \cos{x}-\sin{x} \times \dfrac{d}{dx}\cos{x}}{\cos^2{x}} &\color{red} \text{quotient rule}\\&= \dfrac{\cos{x} \times \cos{x}-\sin{x} \times (-\sin{x})}{\cos^2{x}} \\&= \dfrac{\cos^2{x} + \sin^2{x}}{\cos^2{x}} \\&= […]