Derivatives Of Functions Calculus Examples And Solutions

Calculus, a fundamental branch of mathematics, empowers us to understand change and motion. At its heart lies the concept of the derivative, a powerful tool for determining the instantaneous rate at which a function's output changes with respect to its input. In simpler terms, the derivative unveils the slope of a curve at any given point, providing invaluable insights into the function's behavior.

In this comprehensive guide, we embark on a journey to master the art of finding derivatives for a variety of functions. We'll delve into the core principles of differentiation, explore essential rules and techniques, and apply our knowledge to solve practical problems. Whether you're a student venturing into the realm of calculus or a seasoned mathematician seeking a refresher, this guide will equip you with the knowledge and skills to confidently tackle derivative challenges.

Essential Derivative Rules and Techniques

Before we dive into specific functions, let's establish a solid foundation by understanding the fundamental rules and techniques that govern differentiation. These rules serve as our building blocks, enabling us to find derivatives of complex functions with ease.

The Power Rule

The power rule stands as one of the most frequently used and straightforward rules in differentiation. It dictates that the derivative of a power function, expressed as x raised to the power of n (x^n), is obtained by multiplying the function by the exponent n and reducing the exponent by 1. Mathematically, this can be represented as:

d/dx (x^n) = n * x^(n-1)

For instance, let's consider the function f(x) = x^3. Applying the power rule, we multiply the function by the exponent 3 and reduce the exponent by 1, resulting in the derivative f'(x) = 3x^2.

The Constant Multiple Rule

The constant multiple rule simplifies the differentiation process when dealing with a function multiplied by a constant. It states that the derivative of a constant multiplied by a function is equivalent to the constant multiplied by the derivative of the function. Mathematically, this is expressed as:

d/dx [c * f(x)] = c * f'(x)

For example, if we have the function g(x) = 5x^2, where 5 is the constant and x^2 is the function, we can apply the constant multiple rule. The derivative of g(x) is then g'(x) = 5 * (2x) = 10x.

The Sum and Difference Rule

The sum and difference rule provides a convenient way to find the derivative of a function that is expressed as the sum or difference of multiple terms. It states that the derivative of a sum or difference of terms is equal to the sum or difference of the derivatives of those individual terms. Mathematically, this can be represented as:

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Consider the function h(x) = x^3 + 2x^2 - x. Applying the sum and difference rule, we can differentiate each term separately and then combine the results. The derivative of h(x) is h'(x) = 3x^2 + 4x - 1.

The Product Rule

The product rule comes into play when we need to find the derivative of a function that is expressed as the product of two or more functions. It states that the derivative of the product of two functions, f(x) and g(x), is given by:

d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)

In essence, the product rule tells us to multiply the derivative of the first function by the second function, then add the product of the first function and the derivative of the second function.

Let's illustrate this with an example. Suppose we have the function k(x) = x^2 * sin(x). To find the derivative using the product rule, we first identify f(x) = x^2 and g(x) = sin(x). Then, we find their respective derivatives: f'(x) = 2x and g'(x) = cos(x). Applying the product rule formula, we get k'(x) = (2x * sin(x)) + (x^2 * cos(x)).

The Quotient Rule

The quotient rule is employed when we need to find the derivative of a function that is expressed as the quotient of two functions. It states that the derivative of the quotient of two functions, f(x) and g(x), is given by:

d/dx [f(x) / g(x)] = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]^2

The quotient rule might seem a bit more intricate than the other rules, but it's essential for handling fractions involving functions. To remember it, think of it as (bottom times derivative of top minus top times derivative of bottom) all over (bottom squared).

Let's consider the function l(x) = (x^2 + 1) / (x - 1). To find the derivative using the quotient rule, we first identify f(x) = x^2 + 1 and g(x) = x - 1. Then, we find their derivatives: f'(x) = 2x and g'(x) = 1. Plugging these into the quotient rule formula, we get l'(x) = [(x - 1) * (2x) - (x^2 + 1) * (1)] / (x - 1)^2.

The Chain Rule

The chain rule is a powerful tool for finding the derivatives of composite functions, which are functions nested within other functions. It states that the derivative of a composite function, f(g(x)), is given by:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

In simpler terms, the chain rule tells us to differentiate the outer function f with respect to the inner function g(x), and then multiply the result by the derivative of the inner function g'(x).

Consider the function m(x) = sin(x^2). This is a composite function where the outer function is sin(u) and the inner function is u = x^2. To find the derivative using the chain rule, we first find the derivative of the outer function with respect to u, which is cos(u). Then, we find the derivative of the inner function, which is 2x. Applying the chain rule formula, we get m'(x) = cos(x^2) * 2x.

Applying Derivative Rules to Specific Functions

Now that we have a firm grasp of the essential derivative rules and techniques, let's put our knowledge into practice by finding the derivatives of specific functions. We'll tackle a diverse range of functions, including polynomials, exponential functions, trigonometric functions, and more.

a. $f(x)=3 x^2+5 x-7$

To find the derivative of the function f(x) = 3x^2 + 5x - 7, we'll leverage the power rule, constant multiple rule, and sum/difference rule. This function is a polynomial, and we can differentiate each term individually before combining the results.

Let's break it down step by step:

1. Differentiate the first term, 3x^2: Applying the power rule, we multiply the coefficient 3 by the exponent 2, and then reduce the exponent by 1. This yields 6x.
2. Differentiate the second term, 5x: Again, applying the power rule (remember that x is equivalent to x^1), we multiply the coefficient 5 by the exponent 1, and then reduce the exponent by 1. This gives us 5.
3. Differentiate the third term, -7: The derivative of a constant is always zero.
4. Combine the results: Using the sum and difference rule, we add the derivatives of each term to get the derivative of the entire function. Therefore, the derivative of f(x) is f'(x) = 6x + 5.

b. $g(x)=e^{4 x}-\sin (x)$

In this case, we need to find the derivative of g(x) = e^(4x) - sin(x). This function combines an exponential term and a trigonometric term, so we'll need to employ the chain rule and the derivative of the sine function.

Let's break down the process:

1. Differentiate the first term, e^(4x): This is where the chain rule comes in. We have an outer function, e^u, and an inner function, u = 4x. The derivative of e^u is simply e^u, and the derivative of 4x is 4. Applying the chain rule, we multiply these together to get 4e^(4x).
2. Differentiate the second term, -sin(x): The derivative of sin(x) is cos(x), so the derivative of -sin(x) is -cos(x).
3. Combine the results: Using the sum and difference rule, we subtract the derivative of the second term from the derivative of the first term. This gives us the derivative of g(x) as g'(x) = 4e^(4x) - cos(x).

c. $h(x)=\sec (x)$ Hint: Use quotient rule!

Here, we aim to determine the derivative of h(x) = sec(x). While there's a direct formula for the derivative of sec(x), the hint suggests using the quotient rule, which provides an excellent opportunity to reinforce this fundamental technique. Remember that sec(x) is defined as 1/cos(x), making it a perfect candidate for the quotient rule.

Let's walk through the steps:

1. Rewrite sec(x) as 1/cos(x): This sets up our quotient, where f(x) = 1 (the numerator) and g(x) = cos(x) (the denominator).
2. Find the derivatives of f(x) and g(x): The derivative of f(x) = 1 (a constant) is 0. The derivative of g(x) = cos(x) is -sin(x).
3. Apply the quotient rule formula: Recall that the quotient rule states: d/dx [f(x) / g(x)] = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]^2. Plugging in our values, we get: h'(x) = [cos(x) * 0 - 1 * (-sin(x))] / [cos(x)]^2.
4. Simplify the expression: Simplifying the numerator, we have sin(x). The denominator is cos^2(x). Therefore, h'(x) = sin(x) / cos^2(x).
5. Further simplification (optional): We can rewrite this in terms of sec(x) and tan(x). Remember that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x). Therefore, sin(x) / cos^2(x) can be rewritten as (sin(x)/cos(x)) * (1/cos(x)) = tan(x) * sec(x).

Thus, the derivative of h(x) = sec(x) is h'(x) = sin(x) / cos^2(x), which can also be expressed as h'(x) = tan(x)sec(x).

Conclusion

In this comprehensive guide, we've traversed the landscape of derivatives, unraveling their significance and mastering the techniques for finding them. We began by establishing a solid foundation with essential derivative rules, including the power rule, constant multiple rule, sum and difference rule, product rule, quotient rule, and chain rule. We then applied these rules to a variety of functions, demonstrating their versatility and power.

By understanding and applying these derivative rules and techniques, you've armed yourself with a crucial tool for tackling calculus challenges. The ability to find derivatives opens doors to a deeper understanding of functions, their behavior, and their applications in various fields, from physics and engineering to economics and computer science. As you continue your mathematical journey, remember that practice is key. The more you apply these concepts, the more proficient you'll become in the art of differentiation.