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The subject of this course is \functions of one real variable" so we begin by wondering what a real number \really" is, and then, in the next section, what a function is. 1.
We solve this by using the chain rule and our knowledge of the derivative of ln x. = . There are two shortcuts to differentiating functions involving exponents and logarithms. The four examples above gave. − 2x3 + 5x2 3 − .
We can thus convert the power ax to a power of e: ax = ≥eln(a)¥x = eln(a)x. = d = eln(a)x dxhln(a)xi = eln(a)x ln(a) = ax ln(a). So the derivative of ax is just ax times the constant } ln(a). This is a new rule. = ln(10)10x o 2.302·10x. Also dxh2xi = ln(2)2x o. 0.693 · 2x. Notice how special the base e is: dxhexi = ln(e)ex = 1· ex = ex.
Use the derivative of the natural exponential function, the quotient rule, and the chain rule. y′ = ⎛ ⎝e x2·2⎞ ⎠x·x−1·ex 2 x2 Apply the quotient rule. = ex 2⎛ ⎝2x 2−1⎞ ⎠ x2 Simplify. Find the derivative ofh(x)=xe2x. Example 3.76 Applying the Natural Exponential Function A colony of mosquitoes has an initial population ...
1.1 Derivatives of Logarithmic Functions 1.1.1 The Natural Log - Revisited Recall the natural log function y= ln(x) This is the function whose output is the exponent you raise eto in order to get the value x. For example, ln(e4) = 4. This makes ln(x) the inverse of the exponential function ex. Inverse functions have the special property
We apply the implicit differentiation technique to differentiate logarithmic functions. Example 1. Differentiate y = ln x3 + 1 . Example 2. Example 3. Differentiate f(x) = ln x. Example 4. Differentiate f(x) = log (2 + sin x). Example 5. ln x + 1 p x 2 . Example 6. Find f0(x) if f(x) = ln jxj. The Number e.
In these cases and others, it may be desirable to compute derivatives numerically rather than analytically. The focus of this chapter is numerical differentiation. By the end of this chapter you should be able to derive some basic numerical differentiation schemes and their accuracy.
