Let $f(x) = (9 \sin X + 7 \cos X) \tan^{-1} X$. Find $f^{\prime}(x$\].

Let $f(x) = (9 \sin x + 7 \cos x) \tan^{-1} x$. Find $f^{\prime}(x)$

In this article, we will explore the concept of differentiation and apply it to a given function $f(x) = (9 \sin x + 7 \cos x) \tan^{-1} x$. We will use the product rule and chain rule of differentiation to find the derivative of the function.

The product rule of differentiation states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of the function is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

The chain rule of differentiation states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of the function is given by $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$.

To find the derivative of the function $f(x) = (9 \sin x + 7 \cos x) \tan^{-1} x$, we will use the product rule and chain rule of differentiation.

Let $u(x) = 9 \sin x + 7 \cos x$ and $v(x) = \tan^{-1} x$. Then, we have:

$$f(x) = u(x)v(x)$$

Using the product rule, we get:

$$f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$$

Now, we need to find the derivatives of $u(x)$ and $v(x)$.

Differentiating $u(x)$

To find the derivative of $u(x) = 9 \sin x + 7 \cos x$, we will use the sum rule of differentiation, which states that if we have a function of the form $f(x) = g(x) + h(x)$, then the derivative of the function is given by $f^{\prime}(x) = g^{\prime}(x) + h^{\prime}(x)$.

Using the sum rule, we get:

$$u^{\prime}(x) = 9 \cos x - 7 \sin x$$

Differentiating $v(x)$

To find the derivative of $v(x) = \tan^{-1} x$, we will use the chain rule of differentiation.

Let $g(x) = \tan^{-1} x$. Then, we have:

$$v(x) = g(x)$$

Using the chain rule, we get:

$$v^{\prime}(x) = g^{\prime}(x) \cdot x^{\prime}$$

Now, we need to find the derivative of $g(x)$.

Differentiating $g(x)$

To find the derivative of $g(x) = \tan^{-1} x$, we will use the formula for the derivative of the inverse tangent function, which is given by:

$$\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$$

Using this formula, we get:

$$g^{\prime}(x) = \frac{1}{1 + x^2}$$

Now, we can substitute this expression into the expression for $v^{\prime}(x)$.

Substituting the Derivative of $g(x)$

Substituting the expression for $g^{\prime}(x)$ into the expression for $v^{\prime}(x)$, we get:

$$v^{\prime}(x) = \frac{1}{1 + x^2} \cdot 1$$

Simplifying this expression, we get:

$$v^{\prime}(x) = \frac{1}{1 + x^2}$$

Substituting the Derivatives into the Expression for $f^{\prime}(x)$

Now, we can substitute the expressions for $u^{\prime}(x)$ and $v^{\prime}(x)$ into the expression for $f^{\prime}(x)$.

Substituting the expression for $u^{\prime}(x)$, we get:

$$f^{\prime}(x) = (9 \cos x - 7 \sin x) \tan^{-1} x + (9 \sin x + 7 \cos x) \frac{1}{1 + x^2}$$

Simplifying this expression, we get:

$$f^{\prime}(x) = (9 \cos x - 7 \sin x) \tan^{-1} x + \frac{9 \sin x + 7 \cos x}{1 + x^2}$$

In this article, we used the product rule and chain rule of differentiation to find the derivative of the function $f(x) = (9 \sin x + 7 \cos x) \tan^{-1} x$. We found that the derivative of the function is given by:

$$f^{\prime}(x) = (9 \cos x - 7 \sin x) \tan^{-1} x + \frac{9 \sin x + 7 \cos x}{1 + x^2}$$

$$

Q: What is the derivative of the function $f(x) = (9 \sin x + 7 \cos x) \tan^{-1} x$?

A: The derivative of the function $f(x) = (9 \sin x + 7 \cos x) \tan^{-1} x$ is given by:

$$f^{\prime}(x) = (9 \cos x - 7 \sin x) \tan^{-1} x + \frac{9 \sin x + 7 \cos x}{1 + x^2}$$

Q: How did you find the derivative of the function?

A: To find the derivative of the function, we used the product rule and chain rule of differentiation. We first found the derivatives of the individual functions $u(x) = 9 \sin x + 7 \cos x$ and $v(x) = \tan^{-1} x$, and then substituted these expressions into the expression for $f^{\prime}(x)$.

Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of the function is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of the function is given by $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$.

Q: How do I apply the product rule and chain rule of differentiation?

A: To apply the product rule and chain rule of differentiation, you need to identify the individual functions $u(x)$ and $v(x)$, and then find their derivatives. You can then substitute these expressions into the expression for $f^{\prime}(x)$.

Q: What are some common mistakes to avoid when finding the derivative of a function?

A: Some common mistakes to avoid when finding the derivative of a function include:

• Not identifying the individual functions $u(x)$ and $v(x)$ correctly
• Not finding the derivatives of the individual functions correctly
• Not substituting the expressions for the derivatives into the expression for $f^{\prime}(x)$ correctly
• Not simplifying the expression for $f^{\prime}(x)$ correctly

Q: How can I practice finding the derivative of a function?

A: You can practice finding the derivative of a function by working through examples and exercises in a textbook or online resource. You can also try finding the derivative of a function on your own, and then checking your answer with a calculator or online resource.

Q: What are some real-world applications of finding the derivative of a function?

A: Finding the derivative of a function has many real-world applications, including:

• Modeling population growth and decline
• Modeling the motion of objects
• Finding the maximum and minimum values of a function
• Finding the rate of change of a function

Q: How can I use the derivative of a function to solve a problem?

A: You can use the derivative of a function to solve a problem by identifying the function and its derivative, and then using the derivative to find the rate of change of the function. You can also use the derivative to find the maximum and minimum values of the function.