Question

Compute the derivative of the given function. $$h(x)=x^{1.5}$$

Short answer

The derivative is \( h'(x) = 1.5x^{0.5} \).

Step-by-step solution

Identify the Function

The given function is a power function: \( h(x) = x^{1.5} \).

Recall the Power Rule

The power rule for differentiation states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Here, \( n = 1.5 \).

Apply the Power Rule

Using the power rule, differentiate \( h(x) = x^{1.5} \) to find \( h'(x) \). Multiply the exponent by the coefficient and subtract one from the exponent: \( h'(x) = 1.5x^{1.5 - 1} \).

Simplify the Derivative

Simplify the expression for the derivative: \( h'(x) = 1.5x^{0.5} \).

Key concepts

Power Function

A power function is any function of the form \( f(x) = x^n \) where \( n \) is a real number. Power functions are simple yet fundamental in calculus and algebra. These functions can have positive, negative, or fractional exponents. When dealing with a power function, it's important to understand how the value of \( n \) influences the graph and behavior.

• If \( n \) is positive, the function generally tends to increase as \( x \) increases.
• If \( n \) is negative, the function will typically decrease and might suggest an asymptotic behavior as \( x \) approaches zero.
• If \( n \) is fractional, like in the case with \( x^{1.5} \), the function involves roots. Specifically, \( x^{1.5} \) can be seen as \( x \times x^{0.5} \), or \( x \times \sqrt{x} \).

Understanding power functions is crucial as they form the basis for polynomial and rational functions, which are commonly encountered in calculus.

Power Rule

The power rule is a key differentiation technique in calculus. It provides a quick way to find the derivative of power functions. According to the power rule, if you have a function \( f(x) = x^n \), its derivative \( f'(x) \) is defined as \( nx^{n-1} \). This means you take the exponent, multiply it by the function, then reduce the exponent by one.The simplicity of the power rule makes it a favorite among students for tasks involving derivatives. Keep in mind that the rule applies to both integer and fractional exponents.With our given function \( h(x) = x^{1.5} \), applying the power rule means:

• Identifying the exponent (in this case, \( 1.5 \))
• Multiplying the exponent by the function (\( 1.5 \times x^{1.5} \))
• Subtracting one from the exponent to get \( x^{0.5} \)

This resulting derivative tells us how the function \( h(x) \) changes with respect to \( x \).

Differentiation

Differentiation is a fundamental concept in calculus that helps us find the rate of change of a function. It's like asking, "How does this function behave as the input varies?". By differentiating a function, we obtain a new function that describes the slope of the original function at any given point. This is crucial for numerous real-world applications, including physics and engineering.Differentiation can be applied to various types of functions, but it often requires different rules or techniques depending on the function's form.

• The power rule is one of these methods, particularly for power functions.
• For more complex functions, other rules like product, quotient, or chain rules might be needed.
• Understanding the underlying principles of differentiation makes it easier to work with different types of problems and functional forms.

In the context of our function \( h(x) = x^{1.5} \), differentiation helps us learn how \( h(x) \) grows or shrinks as \( x \) increases, which is often of great interest in various analyses.