Derivative of exponential functions

In Chapter 1, we talked about Limits and we discussed about how important it is to understand the concepts behind this topic. In this chapter, we will talk about the derivative, which is a subject that you will deal with, all throughout your course in Calculus.

In derivation, you need to remember your lessons about the slope of tangent line or the instantaneous rate of change, as this is the main concept applied in this chapter. As we have learned in the previous courses, slope is defined as the ratio between the change in Y and the change in X. In other words, slope is equal to ΔriseΔrun\frac{\Delta rise}{\Delta run}.

It is theorized that we can find the average slope between two points. So, if we want to know the slope between a given unknown point, we need to find the Δ\Delta run or the Δ\Delta x that approaches zero. “The derivative of” is written mathematically as, ddx\frac{d}{dx}.

In this chapter, we will have an 11 part discussion about derivative. For the first three chapters, we will look at the definition of derivative, the power rule and the slope and equation of a tangent line. We need to review about the power rule to help us derive long and complicated polynomial functions with large exponents.

For the next parts of the chapter, we will look at the chain rule. This is used in solving for the derivative of a composite function. We will discuss about the derivative of trigonometric function, exponential function, inverse trigonometric, and logarithmic functions. In order for us to understand these two topics, we will also study the differential rules for trigonometric function, logarithmic function, exponential function, and inverse trig-function (arc function).

We will also discuss about the product rule and quotient rule which are used in simplifying functions that we need to derive. Apart from the topics mentioned above, we will also talk about implicit differentiation. To let us understand the implicit functions, we will also look at the explicit function through the set of exercises we will have.

Derivative of exponential functions

An exponential function is a function containing a numerical base with at least one variable in its exponent. In this section, we will learn how to differentiate exponential functions, including natural exponential functions and other composite functions that require the application of the Chain Rule.


Differential Rules – Exponential Functions

ddxcx=cxlnc\frac{{d}}{{{d}x}}\;{c^x} = {c^x} \cdot \ln c
ddxc()=c()lncddx()\frac{{d}}{{{d}x}}\;{c^{\left( {\;\;\;\;} \right)}} = {c^{\left( {\;\;\;\;} \right)}} \cdot \ln c \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)

ddxex=ex\frac{{d}}{{{d}x}}\;{{e}^x} = {{e}^x}
ddxe()=e()ddx()\frac{{d}}{{{d}x}}\;{{e}^{\left( {\;\;\;\;} \right)}} = {{e}^{\left( {\;\;\;\;} \right)}} \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
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Derivative of exponential functions

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