Question

Differentiate the following funetions: \(u=x^{2} \sqrt{a^{2}-x^{2}}\)

Short answer

Question: Find the derivative of the function \(u = x^{2}\sqrt{a^{2}-x^{2}}\). Answer: \((u)' = 2x\sqrt{a^2 - x^2} - \frac{x^3}{\sqrt{a^2 - x^2}}\).

Step-by-step solution

Finding the derivative of \(u_1(x) = x^2\)

To find the derivative of \(u_1(x) = x^2\), we simply apply the power rule of differentiation, which states that \((x^n)' = nx^{n-1}\) where n is a constant. In this case, \(n = 2\) and the derivative is: \((u_1)' = \frac{d}{dx}(x^2) = 2x^{2-1} = 2x\).

Finding the derivative of \(u_2(x) = \sqrt{a^2 - x^2}\)

To find the derivative of \(u_2(x) = \sqrt{a^2 - x^2}\), we need to apply the chain rule. The outer function is \(f(z) = \sqrt{z}\), and the inner function is \(z(x) = a^2 - x^2\). Let's differentiate the outer and inner functions. - For the outer function: \(f'(z) = \frac{d}{dz}(\sqrt{z}) = \frac{1}{2\sqrt{z}}\). - For the inner function: \(z'(x) = \frac{d}{dx}(a^2 - x^2) = -2x\). Now, applying the chain rule, we have: \((u_2)' = \frac{d}{dx}(u_2) = f'(z(x))\cdot z'(x) = \frac{1}{2\sqrt{z}}\cdot(-2x) = -\frac{x}{\sqrt{a^2-x^2}}\).

Applying the product rule to find the derivative of the entire function

Now that we have the derivatives of both functions, we can apply the product rule to find the derivative of the entire function \(u(x)\): \((u)' = (u_1'u_2 + u_1u_2')= (2x)(\sqrt{a^2 - x^2}) + (x^2)\left(-\frac{x}{\sqrt{a^2 - x^2}}\right)\). Finally, we can simplify the expression: \((u)' = \left(2x\sqrt{a^2 - x^2}\right) - x^3\left(\frac{1}{\sqrt{a^2 - x^2}}\right)\). So, the derivative of the given function is \(\boxed{(u)' = 2x\sqrt{a^2 - x^2} - \frac{x^3}{\sqrt{a^2 - x^2}}}\).

Key concepts

Power Rule

The power rule is a fundamental concept in differentiation that simplifies the process of finding the derivative of power functions. If you have a function of the form \( x^n \), where \( n \) is a constant, the power rule states that the derivative is \( n \cdot x^{n-1} \). This makes it straightforward to work with polynomial functions.
For example, if you need to differentiate \( x^2 \), applying the power rule provides:
\( \frac{d}{dx}(x^2) = 2x^{2-1} = 2x \).
The ability to quickly calculate derivatives using this rule is invaluable, especially when dealing with multiple terms in an expression, as it enables rapid and error-free results. To maximize understanding, always identify the exponent before applying the rule.

Chain Rule

The chain rule is a technique used for differentiating a composite function, which is a function of a function. It is especially useful when dealing with nested functions, where one function is inside another. The chain rule states that the derivative of a composite function \( f(g(x)) \) is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function:

\( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).
In practice, if you need to differentiate \( \sqrt{a^2 - x^2} \), you treat the process like peeling layers of an onion. Here, the outer function is the square root, \( f(z) = \sqrt{z} \), and the inner function is the quadratic, \( z(x) = a^2 - x^2 \). By first differentiating the outer function with respect to \( z \) and then the inner function with respect to \( x \), you apply:

\( f'(z) = \frac{1}{2\sqrt{z}}\) and \( z'(x) = -2x \)
thus making \( (u_2)' = -\frac{x}{\sqrt{a^2-x^2}} \).
When using the chain rule, take care to always differentiate the inner function and multiply it with the derivative of the outer function.

Product Rule

The product rule is crucial when differentiating products of two functions. It allows you to find the derivative of a function that is the product of two simpler functions. According to the product rule, if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product \( uv \) is given by:

\((uv)' = u'v + uv' \).
Suppose you are differentiating \( x^2 \cdot \sqrt{a^2-x^2} \). First, determine the derivatives of each function individually. The derivative of \( u_1(x) = x^2 \) has been found using the power rule as \( 2x \), and the derivative of \( u_2(x) = \sqrt{a^2 - x^2} \) with the chain rule as \( -\frac{x}{\sqrt{a^2 - x^2}} \).
Applying the product rule yields:

\( (u)' = (2x)(\sqrt{a^2 - x^2}) + (x^2)(-\frac{x}{\sqrt{a^2-x^2}}) \).
By substituting and simplifying, the result becomes:

\(2x\sqrt{a^2 - x^2} - \frac{x^3}{\sqrt{a^2-x^2}} \).
Always remember that with the product rule, you need the derivatives of both functions, which are then combined through addition, making sure not to overlook any terms.