Derivatives Of Exponential And Logarithmic Functions

Let's dive into the fascinating world of calculus and explore the derivatives of exponential and logarithmic functions, essential tools for understanding rates of change in various fields, from finance to physics.

Derivatives of Exponential and Logarithmic Functions: A Comprehensive Guide

Exponential and logarithmic functions are fundamental building blocks in mathematics, modeling growth, decay, and many other natural phenomena. Their derivatives unlock a deeper understanding of their behavior, allowing us to analyze how these functions change. This guide will provide a thorough exploration of these derivatives, equipping you with the knowledge and tools to confidently apply them.

Exponential Functions: A Quick Recap

Before we delve into derivatives, let's briefly review exponential functions. An exponential function is defined as:

f(x) = a^x

where 'a' is a constant called the base, and 'x' is the exponent. Key characteristics of exponential functions include:

• When a > 1, the function represents exponential growth.
• When 0 < a < 1, the function represents exponential decay.
• The function always passes through the point (0, 1) since a^0 = 1 for any a ≠ 0.

A particularly important exponential function is the natural exponential function, where the base is the mathematical constant e (approximately 2.71828):

f(x) = e^x

This function plays a crucial role in calculus due to its unique derivative property.

Logarithmic Functions: A Quick Recap

Logarithmic functions are the inverses of exponential functions. The logarithmic function with base 'a' is written as:

f(x) = logₐ(x)

This function answers the question: "To what power must we raise 'a' to get 'x'?" Key characteristics of logarithmic functions include:

• The domain of the function is x > 0.
• The function always passes through the point (1, 0) since logₐ(1) = 0 for any a ≠ 1.
• When a > 1, the function is increasing.
• When 0 < a < 1, the function is decreasing.

Similar to exponential functions, there's a particularly important logarithmic function: the natural logarithm, where the base is e:

f(x) = ln(x) (which is equivalent to logₑ(x))

The natural logarithm is the inverse of the natural exponential function.

The Derivative of the Natural Exponential Function (e^x)

The derivative of the natural exponential function, e^x, is quite remarkable: it is itself!

d/dx (e^x) = e^x

This seemingly simple result has profound implications. It means that the rate of change of e^x at any point is equal to the value of the function at that point. This property makes e^x a fundamental solution to many differential equations modeling growth and decay.

Proof (Using the Limit Definition of the Derivative):

Recall the limit definition of the derivative:

f'(x) = lim (h->0) [f(x + h) - f(x)] / h

For f(x) = e^x, we have:

f'(x) = lim (h->0) [e^(x + h) - e^x] / h

Using the properties of exponents, we can rewrite e^(x + h) as e^x * e^h:

f'(x) = lim (h->0) [e^x * e^h - e^x] / h

Factor out e^x:

f'(x) = e^x * lim (h->0) [e^h - 1] / h

The limit lim (h->0) [e^h - 1] / h is a standard limit that equals 1. Therefore:

f'(x) = e^x * 1 = e^x

This confirms that the derivative of e^x is indeed e^x.

The Chain Rule and the Derivative of e^u(x)

Often, we encounter exponential functions where the exponent is not simply 'x' but a function of 'x', denoted as u(x). In such cases, we use the chain rule to find the derivative:

d/dx [e^u(x)] = e^u(x) * u'(x)

where u'(x) is the derivative of u(x) with respect to x.

Examples:

1. Find the derivative of f(x) = e^(x^2):

   Here, u(x) = x^2, so u'(x) = 2x. Applying the chain rule:

   f'(x) = e^(x^2) * 2x = 2xe^(x^2)

2. Find the derivative of g(x) = e^(sin(x)):

   Here, u(x) = sin(x), so u'(x) = cos(x). Applying the chain rule:

   g'(x) = e^(sin(x)) * cos(x) = cos(x)e^(sin(x))

The Derivative of a^x (where a is a constant and a > 0)

Now, let's consider the derivative of a general exponential function, a^x, where 'a' is a positive constant. We can express a^x in terms of the natural exponential function using the identity:

a^x = e^(ln(a^x)) = e^(x * ln(a))

Now, we can differentiate using the chain rule:

d/dx (a^x) = d/dx [e^(x * ln(a))] = e^(x * ln(a)) * d/dx (x * ln(a))

Since ln(a) is a constant, d/dx (x * ln(a)) = ln(a). Therefore:

d/dx (a^x) = e^(x * ln(a)) * ln(a) = a^x * ln(a)

Summary:

d/dx (a^x) = a^x * ln(a)

Examples:

1. Find the derivative of f(x) = 2^x:

   Applying the formula:

   f'(x) = 2^x * ln(2)

2. Find the derivative of g(x) = 10^x:

   Applying the formula:

   g'(x) = 10^x * ln(10)

The Derivative of the Natural Logarithm (ln(x))

The derivative of the natural logarithm, ln(x), is:

d/dx (ln(x)) = 1/x

This result is another fundamental derivative in calculus. It tells us that the rate of change of ln(x) decreases as x increases.

Proof (Using Implicit Differentiation):

Let y = ln(x). Then, e^y = x. Now, we can differentiate both sides with respect to x using implicit differentiation:

d/dx (e^y) = d/dx (x)

Applying the chain rule to the left side:

e^y * (dy/dx) = 1

Now, solve for dy/dx:

dy/dx = 1 / e^y

Since e^y = x:

dy/dx = 1/x

Therefore, the derivative of ln(x) is 1/x.

The Chain Rule and the Derivative of ln(u(x))

When we have the natural logarithm of a function of x, ln(u(x)), we again use the chain rule:

d/dx [ln(u(x))] = (1 / u(x)) * u'(x) = u'(x) / u(x)

Examples:

1. Find the derivative of f(x) = ln(x^2 + 1):

   Here, u(x) = x^2 + 1, so u'(x) = 2x. Applying the chain rule:

   f'(x) = (2x) / (x^2 + 1)

2. Find the derivative of g(x) = ln(sin(x)):

   Here, u(x) = sin(x), so u'(x) = cos(x). Applying the chain rule:

   g'(x) = cos(x) / sin(x) = cot(x)

The Derivative of logₐ(x) (where a is a constant and a > 0, a ≠ 1)

To find the derivative of a general logarithm, logₐ(x), we can use the change of base formula to express it in terms of the natural logarithm:

logₐ(x) = ln(x) / ln(a)

Since ln(a) is a constant, we can differentiate:

d/dx [logₐ(x)] = d/dx [ln(x) / ln(a)] = (1 / ln(a)) * d/dx [ln(x)]

We know that d/dx [ln(x)] = 1/x. Therefore:

d/dx [logₐ(x)] = (1 / ln(a)) * (1/x) = 1 / (x * ln(a))

Summary:

d/dx [logₐ(x)] = 1 / (x * ln(a))

Examples:

1. Find the derivative of f(x) = log₂(x):

   Applying the formula:

   f'(x) = 1 / (x * ln(2))

2. Find the derivative of g(x) = log₁₀(x):

   Applying the formula:

   g'(x) = 1 / (x * ln(10))

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to differentiate complicated functions involving products, quotients, and powers of functions. It involves taking the natural logarithm of both sides of an equation, simplifying using logarithmic properties, and then differentiating implicitly.

Steps for Logarithmic Differentiation:

1. Take the natural logarithm of both sides: If you have y = f(x), take ln(y) = ln(f(x)).
2. Simplify using logarithmic properties: Use properties like ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), and ln(a^b) = b * ln(a) to simplify the right-hand side.
3. Differentiate both sides implicitly with respect to x: Remember to apply the chain rule on the left side (d/dx ln(y) = (1/y) * dy/dx).
4. Solve for dy/dx: Isolate dy/dx on one side of the equation.
5. Substitute y = f(x): Replace 'y' with the original function f(x) to express the derivative in terms of x.

Examples:

1. Find the derivative of y = x^x:

   • Take the natural logarithm of both sides: ln(y) = ln(x^x) = x * ln(x)
   • Differentiate both sides implicitly: (1/y) * dy/dx = ln(x) + x * (1/x) = ln(x) + 1
   • Solve for dy/dx: dy/dx = y * (ln(x) + 1)
   • Substitute y = x^x: dy/dx = x^x * (ln(x) + 1)

2. Find the derivative of y = (x^2 * sqrt(x + 1)) / ( (x-1)^(2/3) * cos(x) ):

   • Take the natural logarithm of both sides:

     ln(y) = ln( (x^2 * sqrt(x + 1)) / ( (x-1)^(2/3) * cos(x) ) )

   • Simplify using logarithmic properties:

     ln(y) = ln(x^2) + ln(sqrt(x + 1)) - ln( (x-1)^(2/3) ) - ln(cos(x))

     ln(y) = 2ln(x) + (1/2)ln(x + 1) - (2/3)ln(x - 1) - ln(cos(x))

   • Differentiate both sides implicitly:

     (1/y) * dy/dx = (2/x) + (1/2) * (1/(x + 1)) - (2/3) * (1/(x - 1)) - (-sin(x) / cos(x))

     (1/y) * dy/dx = (2/x) + (1/(2(x + 1))) - (2/(3(x - 1))) + tan(x)

   • Solve for dy/dx:

     dy/dx = y * [ (2/x) + (1/(2(x + 1))) - (2/(3(x - 1))) + tan(x) ]

   • Substitute y = (x^2 * sqrt(x + 1)) / ( (x-1)^(2/3) * cos(x) ):

     dy/dx = (x^2 * sqrt(x + 1)) / ( (x-1)^(2/3) * cos(x) ) * [ (2/x) + (1/(2(x + 1))) - (2/(3(x - 1))) + tan(x) ]

Applications of Derivatives of Exponential and Logarithmic Functions

Derivatives of exponential and logarithmic functions have numerous applications in various fields, including:

• Modeling Growth and Decay: Exponential functions are used to model population growth, radioactive decay, and compound interest. Their derivatives help us understand the rate of growth or decay at a given time.
• Optimization Problems: Logarithmic functions are often used in optimization problems, such as finding the maximum or minimum values of a function.
• Related Rates Problems: These derivatives are crucial in related rates problems, where we need to find the rate of change of one quantity in terms of the rate of change of another.
• Curve Sketching: Derivatives help us analyze the shape of exponential and logarithmic functions, including their intervals of increase and decrease, concavity, and inflection points.
• Finance: Derivatives are used extensively in finance for modeling investment growth, calculating present and future values, and understanding risk.
• Physics: Derivatives are used in physics to model various phenomena, such as the motion of objects, the flow of heat, and the behavior of electrical circuits.
• Chemistry: Derivatives are used in chemical kinetics to study reaction rates.

Common Mistakes to Avoid

• Forgetting the Chain Rule: When differentiating composite functions like e^(u(x)) or ln(u(x)), remember to apply the chain rule and multiply by u'(x).
• Incorrectly Applying Logarithmic Properties: Ensure you correctly apply logarithmic properties when using logarithmic differentiation.
• Confusing a^x and x^a: Remember that the derivative of a^x is a^x * ln(a), while the derivative of x^a is a * x^(a-1). These are different rules!
• Forgetting the Constant 'ln(a)': When differentiating a^x or logₐ(x), don't forget to include the ln(a) term.
• Not Simplifying After Logarithmic Differentiation: After implicit differentiation in logarithmic differentiation, remember to isolate dy/dx and substitute back for 'y'.

Practice Problems

To solidify your understanding, try these practice problems:

1. Find the derivative of f(x) = e^(3x^2 + 2x - 1)
2. Find the derivative of g(x) = ln(cos(x) + x)
3. Find the derivative of h(x) = 5^(sin(x))
4. Find the derivative of y = x^(sin(x)) using logarithmic differentiation.
5. Find the derivative of k(x) = log₃(x^2 + 1)

(Solutions are provided at the end of this article)

Conclusion

Mastering the derivatives of exponential and logarithmic functions is crucial for success in calculus and its applications. By understanding the fundamental rules, the chain rule, and techniques like logarithmic differentiation, you can confidently tackle a wide range of problems involving these powerful functions. Remember to practice regularly and pay attention to detail to avoid common mistakes. This knowledge will serve as a solid foundation for further exploration of calculus and its applications in various fields.

Solutions to Practice Problems:

1. f'(x) = (6x + 2) * e^(3x^2 + 2x - 1)
2. g'(x) = (1 - sin(x)) / (cos(x) + x)
3. h'(x) = cos(x) * 5^(sin(x)) * ln(5)
4. dy/dx = x^(sin(x)) * [ (sin(x)/x) + ln(x)cos(x) ]
5. k'(x) = (2x) / [(x^2 + 1) * ln(3)]