Let F ( X ) = X 3 2 F(x) = X^{\frac{3}{2}} F ( X ) = X 2 3 ​ .Find F ′ ( X F^{\prime}(x F ′ ( X ].

Let $f(x) = x^{\frac{3}{2}}$. Find $f^{\prime}(x)$

In this article, we will explore the concept of finding the derivative of a function, specifically the function $f(x) = x^{\frac{3}{2}}$. The derivative of a function represents the rate of change of the function with respect to its input variable. In this case, we will use the power rule of differentiation to find the derivative of $f(x)$.

The power rule of differentiation states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. This rule can be applied to any function of the form $f(x) = x^n$, where $n$ is a real number.

Applying the Power Rule to $f(x) = x^{\frac{3}{2}}$

To find the derivative of $f(x) = x^{\frac{3}{2}}$, we can apply the power rule of differentiation. Using the power rule, we have:

$$f^{\prime}(x) = \frac{3}{2}x^{\frac{3}{2}-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = \frac{3}{2}x^{\frac{1}{2}}$$

The derivative $f^{\prime}(x) = \frac{3}{2}x^{\frac{1}{2}}$ can be simplified further. We can rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$, so the derivative becomes:

$$f^{\prime}(x) = \frac{3}{2}\sqrt{x}$$

The derivative $f^{\prime}(x) = \frac{3}{2}\sqrt{x}$ represents the rate of change of the function $f(x) = x^{\frac{3}{2}}$ with respect to $x$. This means that the derivative gives us the slope of the tangent line to the graph of $f(x)$ at any point $x$.

To visualize the derivative, we can graph the function $f(x) = x^{\frac{3}{2}}$ and its derivative $f^{\prime}(x) = \frac{3}{2}\sqrt{x}$. The graph of $f(x)$ is a curve that opens upwards, and the graph of $f^{\prime}(x)$ is a curve that opens upwards as well.

In this article, we found the derivative of the function $f(x) = x^{\frac{3}{2}}$ using the power rule of differentiation. The derivative is given by $f^{\prime}(x) = \frac{3}{2}\sqrt{x}$. This represents the rate of change of the function with respect to $x$, and it can be used to find the slope of the tangent line to the graph of $f(x)$ at any point $x$.

1. Find the derivative of the function $f(x) = x^4$.
2. Find the derivative of the function $f(x) = x^{\frac{5}{3}}$.
3. Find the derivative of the function $f(x) = x^2$.

1. Using the power rule, we have:

$$f^{\prime}(x) = 4x^{4-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = 4x^3$$

2. Using the power rule, we have:

$$f^{\prime}(x) = \frac{5}{3}x^{\frac{5}{3}-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = \frac{5}{3}x^{\frac{2}{3}}$$

3. Using the power rule, we have:

$$f^{\prime}(x) = 2x^{2-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = 2x$$

$$$$

In our previous article, we explored the concept of finding the derivative of a function using the power rule of differentiation. In this article, we will answer some frequently asked questions about finding derivatives.

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. This rule can be applied to any function of the form $f(x) = x^n$, where $n$ is a real number.

Q: How do I apply the power rule to find the derivative of a function?

A: To apply the power rule, simply multiply the exponent of the function by the coefficient of the function, and then subtract 1 from the exponent. For example, if we have the function $f(x) = 3x^4$, we can apply the power rule as follows:

$$f^{\prime}(x) = 3(4)x^{4-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = 12x^3$$

Q: What if the function has a negative exponent?

A: If the function has a negative exponent, we can apply the power rule as follows:

$$f^{\prime}(x) = -n x^{-n-1}$$

For example, if we have the function $f(x) = -2x^{-3}$, we can apply the power rule as follows:

$$f^{\prime}(x) = -(-3) (-2) x^{-3-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = 6x^{-4}$$

Q: What if the function has a fractional exponent?

A: If the function has a fractional exponent, we can apply the power rule as follows:

$$f^{\prime}(x) = \frac{n}{2} x^{\frac{n}{2}-1}$$

For example, if we have the function $f(x) = \frac{3}{2}x^{\frac{3}{2}}$, we can apply the power rule as follows:

$$f^{\prime}(x) = \frac{3}{2} \left(\frac{3}{2}\right) x^{\frac{3}{2}-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = \frac{9}{4}x^{\frac{1}{2}}$$

Q: Can I use the power rule to find the derivative of a function with a variable exponent?

A: Unfortunately, the power rule can only be applied to functions with constant exponents. If the exponent is a variable, we will need to use other techniques, such as implicit differentiation or the chain rule, to find the derivative.

Q: What if I have a function with multiple terms?

A: If you have a function with multiple terms, you can apply the power rule to each term separately. For example, if we have the function $f(x) = 2x^3 + 3x^2$, we can apply the power rule as follows:

$$f^{\prime}(x) = 2(3)x^{3-1} + 3(2)x^{2-1}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = 6x^2 + 6x$$

In this article, we answered some frequently asked questions about finding derivatives using the power rule of differentiation. We hope that this article has been helpful in clarifying any confusion you may have had about finding derivatives. If you have any further questions, please don't hesitate to ask.