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Transcript
Basic Calculus
Basic Calculus
Calculus is a branch of mathematics that helps us understand changes between values that are related by a function. ... Calculus is used in many different areas such as physics, astronomy, biology, engineering, economics, medicine and sociology.
Differentiation
AVERAGE AND INSTANTANEOUS RATE OF CHANGE
DERIVATIVE OF TRIGO FUNCTIONS
process of finding the derivative of a function.-Derivative − the derivative of f at x is given by lim ∆x→0 f(x+∆x)−f(x) ∆x
Derivatives of Basic Trigonometric Functions They are as follows: (sinx)′=cosx,(cosx)′=−sinx. Using the quotient rule it is easy to obtain an expression for the derivative
“Average Rate of Change” of f(x) with respect to x for a function as x changes from X1to X2. - tells how much the y − values of a function change versus how much the x − values change between two particular points on that function.
DERIVATIVE OF EXPONENTIAL AND LOGARITHMIC FUNCTION
The next set of functions that we want to take a look at are exponential and logarithm functions. The most common exponential and logarithm functions in a calculus course are the natural exponential function, e x , and the natural logarithm function, ln ( x ) . We will take a more general approach however and look at the general exponential and logarithm function.
Exponential Functions - We’ll start off by looking at the exponential function, f ( x ) = a x We want to differentiate this. The power rule that we looked at a couple of sections ago won’t work as that required the exponent to be a fixed number and the base to be a variable. That is exactly the opposite from what we’ve got with this function. So, we’re going to have to start with the definition of the derivative.
Derivative of Logarithmic Functions- On the page Definition of the Derivative, we have found the expression for the derivative of the natural logarithm function y=lnx: (lnx)′=1x. Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative. Δy=loga(x+Δx)−logax.