Chapter 3: Problem 72
Find the following higher-order derivatives. $$\frac{d^{n}}{d x^{n}}\left(2^{x}\right)$$
Short Answer
Expert verified
Question: Find a general expression for the nth order derivative of $$2^x$$ with respect to x.
Answer: $$\frac{d^{n}}{dx^{n}}(2^x) = (\ln 2)^n(2^x)$$
Step by step solution
01
Find the first derivative
To start, we need to find the first derivative of $$2^x$$. To do this, we can use the formula: $$\frac{d}{dx}(a^x) = (\ln a)a^x$$ (where a is the base). In our case, a = 2.
So, the first derivative is:
$$\frac{d}{dx} (2^x) = (\ln 2)2^x$$
02
Find the second derivative
Next, we need to find the second derivative of $$2^x$$. To do this, we can differentiate the first derivative with respect to x.
So, the second derivative is:
$$\frac{d^2}{dx^2} (2^x) = \frac{d}{dx} (\ln 2)(2^x) = (\ln 2)^2(2^x)$$
03
Find the nth derivative
We can observe a pattern in the above two steps. Each derivative with respect to x is just the previous derivative multiplied by the $$\ln 2$$ factor. So, we can generalize this pattern to the nth derivative:
$$\frac{d^{n}}{dx^{n}}(2^x) = (\ln 2)^n(2^x)$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
Exponential functions are one of the fundamental concepts in mathematics, particularly in calculus. They are functions of the form
In our exercise, the base
f(x) = a^x
, where a
is a positive constant called the base, and x
is the exponent. These functions exhibit growth or decay at a constant percentage rate, making them incredibly useful for modeling real-world phenomena such as population growth, radioactive decay, and interest rates. In our exercise, the base
a
is 2
, and we are interested in the behavior of 2^x
as x
increases. Understanding how to work with these functions and differentiate them is critical in various fields of science and engineering. An interesting property of these functions is that their derivative is proportional to the function itself, a characteristic that simplifies the process of taking higher-order derivatives. Chain Rule
The chain rule is a powerful derivative rule that allows us to differentiate composite functions, where one function is applied inside another. To put it simply, if given a function
When we derive exponential functions like
h(x) = f(g(x))
, the derivative h'(x)
is found by multiplying the derivative of the outside function f
evaluated at g(x)
by the derivative of the inside function g
. This can be captured in the formula h'(x) = f'(g(x)) ยท g'(x)
. When we derive exponential functions like
2^x
, the chain rule isn't directly used since x
isn't a composition of functions. However, understanding the chain rule is key when dealing with more complex functions involving exponentials, as it often comes into play when the exponent itself is a function of x
. Natural Logarithm
The natural logarithm, usually written as
For any exponential function
ln(x)
, is the logarithm to the base e
, where e
is the Euler's number, approximately equal to 2.71828. It is a special logarithm because of its unique properties in calculus, such as the derivative of ln(x)
being 1/x
. The natural logarithm plays a pivotal role in differentiating exponential functions with bases other than e
, as seen in our exercise. For any exponential function
a^x
, the derivative can be found using the natural logarithm of the base a
: d/dx (a^x) = (ln a)a^x
. This formula underpins finding the higher-order derivatives of 2^x
, since each differentiation multiplies the function by an additional factor of ln 2
. Derivative Rules
Derivative rules are formulas that provide a method to find the derivative of functions. Different rules are used based on the type of function, such as the power rule for polynomial functions, the product rule when functions are multiplied, and the quotient rule for divisions.
In the case of higher-order derivatives of exponential functions, we repeatedly apply the rule for derivatives of exponential functions. For the first derivative of
In the case of higher-order derivatives of exponential functions, we repeatedly apply the rule for derivatives of exponential functions. For the first derivative of
2^x
, we use the fact that d/dx (a^x) = (ln a)a^x
. For each subsequent derivative, we multiply the previous derivative by ln 2
because the exponential function remains the same form with each differentiation. This consistent pattern greatly simplifies the process, allowing us to express the n
th derivative as (ln 2)^n(2^x)
without directly calculating each intermediate step.