 Maxima and Minima with Trigonometric Functions

Home > Differentiation 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 $f'(x)$ to obtain the gratest function. Step 2: [...] Rates of Change

Home > Differentiation Assume $x(t)$ is a displacement, then $x'(t)$ or $\dfrac{dx}{dt}$ is the instantaneous rate of change in displacement with respect to time, which is velocity. Example where quantities vary with time, or with respect to some other value. temperature changes height of the surface of water in a pond speed of a car [...] Velocity and Acceleration

Home > Differentiation 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 $a(t)=v'(t)=x''(t)$ $x(0)$, $v(0)$ and $a(0)$ give the position, velocity [...] Inflection Points (Points of Inflection)

Home > Differentiation Horizontal (stationary) point of inflection (inflection point) If $x \lt a$, then $f'(x) \gt 0$. If $x = a$, then $f'(x) = 0$ and $f''(x) = 0$. If $x \gt a$, then $f'(x) \gt 0$. That is, the gradient is positive either side of the stationary point. If $x \lt a$, then [...] Increasing Functions and Decreasing Functions

Home > Differentiation 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 [...] Finding the Normal Equation

Home > Differentiation 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 [...] Finding the Equation of the Tangent Line

Home > Differentiation 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 [...] Home > Differentiation Given a function $f(x)$, the derivative $f'(x)$ is known as the first derivative. The second derivative of $f(x)$ is the derivative of $f'(x)$, which is $f''(x)$ or the derivative of the first derivative. \displaystyle \begin{align} f'(x) &= \dfrac{d}{dx}f(x) \\ f''(x) &= \dfrac{d}{dx}f'(x) \\ f^{(3)}(x) &= \dfrac{d}{dx}f''(x) \\ f^{(4)}(x) &= \dfrac{d}{dx}f^{(3)}(x) \\ [...] Derivative of Trigonometric Functions Home > Differentiation \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}. Show Solution \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} [...] Derivative of Logarithmic Functions Home > Differentiation \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \dfrac{d}{dx}\log_e{f(x)} = \dfrac{1}{f(x)} \times f'(x) Example 1 Find \displaystyle \dfrac{dy}{dx} if y=\log_e{(x^2+1)}. Show Solution \( \begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{(x^2+1)} \\ &= \dfrac{1}{x^2+1} \times \dfrac{d}{dx}(x^2+1) \\ &= \dfrac{1}{x^2+1} \times 2x \\ &= \dfrac{2x}{x^2+1} \\ \end{align} Example 2 Find \displaystyle \dfrac{dy}{dx} if y=x^2\log_e{(2x-1)}. Show Solution \( \begin{align} \displaystyle \require{color} [...] Derivative of Exponential Functions Home > Differentiation The functions e^{-x}, e^{3x+2} and e^{x^2+2x-1} are all of the form e^{f(x)}. e^{f(x)} \gt 0 for all x, no matter what the function f(x).\displaystyle \dfrac{d}{dx}e^x = e^x\displaystyle \dfrac{d}{dx}e^{f(x)} = e^{f(x)} \times f'(x)Example 1 Find \displaystyle \dfrac{dy}{dx} if y=e^{4x}. Show Solution \( \begin{align} \displaystyle \dfrac{dy}{dx} &= e^{4x} \times \dfrac{d}{dx}4x \\ [...] Quotient Rule of Differentiation Home > Differentiation The quotient rule is a formula for taking the derivative of a quotient of two functions. This formula makes it somewhat easier to keep track of all of the terms. If u(x) and v(x) are two functions of x and \displaystyle f(x)=\dfrac{u(x)}{v(x)}, thenf'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$$Expressions like \displaystyle \dfrac{x^2+x+1}{4x-2}, \displaystyle \dfrac{\sqrt{x+2}}{x^2-4} and \displaystyle [...] Product Rule Differentiation Home > Differentiation The product rule differentiation is used in differential calculus to help calculating the derivative of products of functions. The formula for the product rule differentiation is written for the product of two or more functions. If u(x) and v(x) are two functions of x and f(x)=u(x)v(x), then$$f'(x) = u'(x)v(x) + u(x)v'(x) [...] Chain Rule of Differentiation

Home > Differentiation In differential calculus, the chain rule is a formula for determining the derivative of the combined two or more functions. the chain rule could be used in Leibniz's notation in the following way. If $y=g(u)$ where $u=f(x)$ then $\displaystyle \dfrac{dy}{dx}=\dfrac{dy}{du} \times \dfrac{du}{dx}$. Generally, the chain rule is described as following in its [...] Basic Differentiation Rules

Home > Differentiation Differentiation is the process of finding a derivative or gradient function. Given a function $f(x)$, we obtain $f'(x)$ by differentiating with respect to the variable $x$. There are a number of rules associated with differentiation. These rules can be used to differentiate more complicated functions without having to resort to the sometimes [...] First Principles at a Given $x$-value

Home > Differentiation First Principles at a Given x-Value Suppose we are given a function $f(x)$ and asked to find its derivative at the point where $x=a$. This is actually the gradient of the tangent to the curve at $x=a$, which we write as $f'(a)$. There are two methods for finding $f'(a)$ using first principles: [...]