Tag Archives: Differentiation

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) \\ […]

Derivative of Trigonometric Functions

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}} \\ […]