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**Week 3: Derivative of logarithmic Exponential and Trigonometric Functions. ( Exercise 3.2 , Exercise 3.3)**

On the page Definition of the **Derivative**, we have found the expression for the **derivative** of the natural **logarithm function** y=**ln**x: (**ln**x)′=1x. Now we consider the **logarithmic function** with arbitrary base and obtain a formula for its **derivative**. Δy=**log**a(x+Δx)−**log**ax. Note that the **exponential function** f( x) = **e** x has the special property that its **derivative** is the **function** itself, f′( x) = **e** x = f( x).

In mathematics, the **trigonometric functions** (also called **circular functions**, **angle functions** or **goniometric functions**[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

The trigonometric functions most widely used in modern mathematics are the **sine**, the **cosine**, and the **tangent**. Their reciprocals are respectively the **cosecant**, the **secant**, and the **cotangent**, which are less used.