# Tag Archives: Differentiation # Differentiation and Displacement, Velocity and Acceleration

Distance Distance is the magnitude of the total movement from the start point or a fixed point. Displacement The displacement of a moving position relative to a fixed point. Displacement gives both the distance and direction that a particle is from a fixed point. For example, a particle moves $5$ units forwards from […] # Maxima and Minima with Trigonometric Functions

Periodic motions can be modelled by a trigonometric equation. By differentiating these functions we are then able to solve problems relating to maxima (maximums) and minima (minimums). Remember that the following steps are used when solving a maximum or minimum problem. Step 1: Find $f^{\prime}(x)$ to obtain the gratest function. Step 2: Solve for $x$ […] # Velocity and Acceleration

If a particle $P$ moves in a straight line and its position is given by the displacement function $x(t)$, then: the velocity of $P$ at time $t$ is given by $v(t) = x'(t)$ the acceleration of $P$ at time $t$ is given by $a(t)=v'(t)=x^{\prime \prime}(t)$ $x(0)$, $v(0)$ and $a(0)$ give the position, velocity and acceleration […] # Inflection Points (Points of Inflection)

Horizontal (stationary) point of inflection (inflection point) If $x \lt a$, then $f'(x) \gt 0$ and $f^{\prime \prime}(x) \le 0 \rightarrow$ concave down. If $x = a$, then $f'(x) = 0$ and $f^{\prime \prime}(x) = 0 \rightarrow$ horizontal point inflection. If $x \gt a$, then $f'(x) \gt 0$ and $f^{\prime \prime}(x) \ge 0 \rightarrow$ concave […] A turning point of a function is a point 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. $$\begin{array}{|c|c|c|} \hline f^{\prime}(x) \gt 0 & f'(x) = 0 & f'(x) \lt 0 \\ \hline & \text{maximum} & \\ […] # Increasing Functions and Decreasing Functions Increasing and Decreasing We can determine intervals where a curve is increasing or decreasing by considering f'(x) on the interval in question. f'(x) \gt 0: f(x) is increasing f'(x) \lt 0: f(x) is decreasing Monotone (Monotonic) Increasing or Decreasing Many functions are either increasing or decreasing for all x \in \mathbb{R}. These functions are called […] # Finding the Normal Equations A normal to a curve is a straight line passing through the point where the tangent touches the curve and is perpendicular (at right angles) to the tangent at that point. The gradient of the tangent to a curve is m, then the gradient of the normal is \displaystyle -\dfrac{1}{m}, as the product of the […] # Finding Equations of Tangent Line Consider a curve y=f(x). A tangent to a curve is a straight line which touches the curve at a given point and represents the gradient of the curve at that point.If A is the point with x-coordinate a, then the gradient of the tangent line to the curve at this point is f'(a). The equation […] # Higher Derivatives Given a function f(x), the derivative f^{\prime}(x) is known as the first derivative. The second derivative of f(x) is the derivative of f^{\prime}(x), which is f^{\prime \prime}(x) or the derivative of the first derivative.$$ \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) \\ […] # Derivative of Trigonometric Functions

\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{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}} \\ […]