Rates of Change

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 [...]
Higher Derivatives

Higher Derivatives

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

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

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

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

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 $\displaystyle f(x)=\dfrac{u(x)}{v(x)}$, then $$f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$$ Expressions like $\displaystyle \dfrac{x^2+x+1}{4x-2}$, $\displaystyle \dfrac{\sqrt{x+2}}{x^2-4}$ and $\displaystyle [...]
Chain Rule of Differentiation

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 [...]