# 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 [...]

# Motion Kinematics

Home > Differentiation Displacement Suppose an object $P$ moves along a straight line so that its position $s$ from an origin $O$ is given as some function of time $t$. We write $x=x(t)$ where $t \ge 0$. $x(t)$ is a displacement function and for any value of $t$ it gives the displacement from the origin. [...]

# 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 [...]

# 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 \\ [...]