Question

Using the general rule, differentiate the following with respect to \(x\) : (a) \(y=5 x^{7}\) (b) \(y=3 \sqrt{x}\) (c) \(y=\frac{4}{x^{2}}\)

Short answer

(a) \(35x^6\), (b) \(\frac{3}{2}x^{-1/2}\), (c) \(-8x^{-3}\)

Step-by-step solution

Differentiate the power function in part (a)

The function given is \( y = 5x^7 \). We use the power rule of differentiation which states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \). Applying this to \( y = 5x^7 \), we differentiate:\( \frac{dy}{dx} = 7 \times 5x^{7-1} = 35x^6 \).

Rewrite and differentiate the square root function in part (b)

The function given is \( y = 3\sqrt{x} \). First, rewrite the square root in terms of powers: \( y = 3x^{1/2} \). Now apply the power rule: \( \frac{dy}{dx} = \frac{1}{2} \times 3x^{(1/2)-1} = \frac{3}{2}x^{-1/2} \).

Rewrite and differentiate the rational function in part (c)

The function given is \( y = \frac{4}{x^2} \). Rewrite this as a power function: \( y = 4x^{-2} \). Now apply the power rule: \( \frac{dy}{dx} = -2 \times 4x^{-2-1} = -8x^{-3} \).

Key concepts

Power Rule

The power rule is a fundamental concept in calculus used to find the derivative of functions in the form of a power. This rule states that if you have a function of the form \( y = x^n \), where \( n \) is any real number, then the derivative with respect to \( x \) is \( \frac{dy}{dx} = nx^{n-1} \). This simplifies the differentiation process and is especially helpful for polynomial functions.

In the context of the exercise, let's take example (a) where we have \( y = 5x^7 \). By applying the power rule, you multiply the exponent (7) by the coefficient (5) and decrease the exponent by one. Thus, you obtain \( \frac{dy}{dx} = 7 \times 5 x^{7-1} = 35x^6 \).

It's crucial to remember that the power rule helps simplify the process of finding derivatives for functions that look like "powers of x". It becomes a straightforward exercise of multiplying and reducing exponents.

Chain Rule

Although the provided exercises do not explicitly require the chain rule, it's vital to understand how it interacts with functions composed of other functions—a concept often tied to differentiation. The chain rule is used when a function is "nested" within another. If you have a function \( u(x) \) wrapped inside another function \( f(u) \), to find the derivative \( \frac{dy}{dx} \), you can use the chain rule:

\(\frac{dy}{dx} = \frac{df}{du} \times \frac{du}{dx}\)

In simpler terms, differentiate the outer function (while leaving the inside function alone) and then multiply it by the derivative of the inside function. This rule is especially useful when dealing with complex compositions such as \( y = (3x+5)^4 \), and you would differentiate it accordingly.

Recognizing when to use the chain rule can save you from making errors in more intricate problems involving nested functions.

Rational Functions

Rational functions are expressions of the form \( \frac{p(x)}{q(x)} \), where both the numerator and the denominator are polynomials. Differentiating rational functions often involves simplifying them into forms where you can apply the power rule easily.

For example, consider example (c) from the exercise where the function is \( y = \frac{4}{x^2} \). This is a rational function, and to differentiate it, you can rewrite it as \( y = 4x^{-2} \). Now the power rule can be applied directly to get \( \frac{dy}{dx} = -2 \times 4x^{-3} = -8x^{-3} \).

The process here involves rewriting the rational function into a power of \( x \), making it easier to differentiate. This method turns complex rational expressions into straightforward power functions that are easier to handle, making the differentiation process more efficient.