In this page, we will cover the topics of differential calculus which includes the derivative of a function, its applications etc. Let us find the topics below.

Topics of Differential Calculus |

- Derivative: Definition, Properties, and Formulas
- Proofs of Derivative Formulas (by definition)
- Derivative implies continuity but the converse is not true
- Mod x is continuous at x=0 but not differentiable at x=0

Chain Rule of Derivatives: |

For two functions f(x) and g(x), by the chain rule, the derivative of the composite function f(g(x)) is given by the following formula:

[f(g(x))]$’$ = f$’$(g(x)) ⋅ g$’$(x)

where $’$ denotes the first order derivative.

Examples of Derivatives by Chain Rule:

√sinx | √cosx | √e^{x} |

cos(e^{x}) | cos2x | cos^{4}(x) |

sin^{5}x | $e^{x^2}$ | tan x |

1/(1+x) | tan^{2}x | sin^{3}x |

1/lnx | √2x | √3x |

√logx | e^{1/x} | cos(x^{4}) |

1/logx |

Derivative by First Principles: |

The derivative of a function f(x) by first principle is equal to the limit

f$’$(x) = lim_{h→0} $\dfrac{f(x+h)-f(x)}{h}$.

**For example,** the derivative of x by first principle can be computed as follows:

(x)$’$ = lim_{h→0} $\dfrac{(x+h)- x}{h}$ = lim_{h→0} $\dfrac{h}{h}$ = lim_{h→0} 1 = 1.

Derivative of tanx: The derivative tanx is sec^{2}x.

Derivative of e^{2x}: The derivative of e^{2x} is 2e^{2x}.

Derivative of tanx: The derivative tanx is sec^{2}x.

Derivative of e^{2x}: The derivative of e^{2x} is 2e^{2x}.

Derivative of root x: The derivative of square root of x is 1/(2√x).

Derivative of root logx: The derivative of $\sqrt{\log x}$ is

Derivative of root sinx: The derivative of $\sqrt{\sin x}$ is

Derivative of root cosx: The derivative of $\sqrt{\cos x}$ is

Derivative of root e^{x}: The derivative of $\sqrt{e^x}$ is

Derivative of root(x)+1/root(x): The derivative of $\sqrt{x}+\frac{1}{\sqrt{x}}$ is

Derivative of log3x: The derivative of log3x is 1/x.