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 | √ex |
| cos(ex) | cos2x | cos4(x) |
| sin5x | $e^{x^2}$ | tan x |
| 1/(1+x) | tan2x | sin3x |
| 1/lnx | √2x | √3x |
| √logx | e1/x | cos(x4) |
| 1/logx |
| Derivative by First Principles: |
The derivative of a function f(x) by first principle is equal to the limit
f$’$(x) = limh→0 $\dfrac{f(x+h)-f(x)}{h}$.
For example, the derivative of x by first principle can be computed as follows:
(x)$’$ = limh→0 $\dfrac{(x+h)- x}{h}$ = limh→0 $\dfrac{h}{h}$ = limh→0 1 = 1.
Derivative of tanx: The derivative tanx is sec2x.
Derivative of e2x: The derivative of e2x is 2e2x.
Derivative of tanx: The derivative tanx is sec2x.
Derivative of e2x: The derivative of e2x is 2e2x.
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 ex: 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.