Differentiate The Following Function:$\[ F(x) = 3 \sec X + 2 E^x \tan X \\]$\[ F^{\prime}(x) = \\]$\[\square\\]

In this article, we will focus on differentiating a given function that involves trigonometric and exponential functions. The function is given as $f(x) = 3 \sec x + 2 e^x \tan x$. We will use various differentiation rules to find the derivative of this function.

Understanding the Function

The given function is a combination of two functions: $3 \sec x$ and $2 e^x \tan x$. The first function involves the secant function, which is the reciprocal of the cosine function. The second function involves the exponential function and the tangent function.

Differentiating the Function

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

Step 1: Differentiate the First Term

The first term in the function is $3 \sec x$. To differentiate this term, we will use the chain rule. The derivative of the secant function is given by $\frac{d}{dx} \sec x = \sec x \tan x$. Therefore, the derivative of the first term is $3 \sec x \tan x$.

Step 2: Differentiate the Second Term

The second term in the function is $2 e^x \tan x$. To differentiate this term, we will use the product rule. The derivative of the exponential function is given by $\frac{d}{dx} e^x = e^x$. The derivative of the tangent function is given by $\frac{d}{dx} \tan x = \sec^2 x$. Therefore, the derivative of the second term is $2 e^x \sec^2 x + 2 e^x \tan x$.

Combining the Derivatives

Now that we have differentiated the two terms, we can combine the derivatives to find the derivative of the given function. The derivative of the first term is $3 \sec x \tan x$, and the derivative of the second term is $2 e^x \sec^2 x + 2 e^x \tan x$. Therefore, the derivative of the given function is $f^{\prime}(x) = 3 \sec x \tan x + 2 e^x \sec^2 x + 2 e^x \tan x$.

Simplifying the Derivative

We can simplify the derivative by combining like terms. The derivative can be written as $f^{\prime}(x) = 3 \sec x \tan x + 2 e^x (\sec^2 x + \tan x)$.

Conclusion

In this article, we differentiated a given function that involves trigonometric and exponential functions. We used the product rule and the chain rule to find the derivative of the function. The derivative of the function is $f^{\prime}(x) = 3 \sec x \tan x + 2 e^x (\sec^2 x + \tan x)$.

Key Takeaways

• The product rule states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.
• The chain rule states that if we have a function of the form $f(x) = u(g(x))$, then the derivative is given by $f^{\prime}(x) = u^{\prime}(g(x)) \cdot g^{\prime}(x)$.
• The derivative of the secant function is given by $\frac{d}{dx} \sec x = \sec x \tan x$.
• The derivative of the tangent function is given by $\frac{d}{dx} \tan x = \sec^2 x$.
• The derivative of the exponential function is given by $\frac{d}{dx} e^x = e^x$.

Final Answer

In this article, we will continue to explore the concept of differentiating the given function $f(x) = 3 \sec x + 2 e^x \tan x$. We will answer some frequently asked questions related to this topic.

Q: What is the derivative of the secant function?

A: The derivative of the secant function is given by $\frac{d}{dx} \sec x = \sec x \tan x$.

Q: What is the derivative of the tangent function?

A: The derivative of the tangent function is given by $\frac{d}{dx} \tan x = \sec^2 x$.

Q: What is the derivative of the exponential function?

A: The derivative of the exponential function is given by $\frac{d}{dx} e^x = e^x$.

Q: How do I apply the product rule to differentiate the given function?

A: To apply the product rule, we need to identify the two functions that are being multiplied together. In this case, the two functions are $3 \sec x$ and $2 e^x \tan x$. We then take the derivative of each function separately and multiply them together.

Q: How do I simplify the derivative of the given function?

A: To simplify the derivative, we can combine like terms. In this case, we can combine the terms $3 \sec x \tan x$ and $2 e^x (\sec^2 x + \tan x)$ to get the simplified derivative.

Q: What is the final answer to the problem?

A: The final answer to the problem is $f^{\prime}(x) = 3 \sec x \tan x + 2 e^x (\sec^2 x + \tan x)$.

Q: Can I use the chain rule to differentiate the given function?

A: Yes, you can use the chain rule to differentiate the given function. The chain rule states that if we have a function of the form $f(x) = u(g(x))$, then the derivative is given by $f^{\prime}(x) = u^{\prime}(g(x)) \cdot g^{\prime}(x)$.

Q: How do I apply the chain rule to differentiate the given function?

A: To apply the chain rule, we need to identify the inner function $g(x)$ and the outer function $u(x)$. In this case, the inner function is $g(x) = \sec x$ and the outer function is $u(x) = 3 + 2 e^x \tan x$. We then take the derivative of the inner function and multiply it by the derivative of the outer function.

Q: What are some common mistakes to avoid when differentiating the given function?

A: Some common mistakes to avoid when differentiating the given function include:

• Forgetting to apply the product rule when differentiating the second term.
• Forgetting to simplify the derivative after applying the product rule.
• Not using the chain rule when differentiating the secant function.
• Not using the chain rule when differentiating the tangent function.

Conclusion

In this article, we have answered some frequently asked questions related to differentiating the given function $f(x) = 3 \sec x + 2 e^x \tan x$. We have also provided some tips and tricks for avoiding common mistakes when differentiating this function.

Key Takeaways

• The derivative of the secant function is given by $\frac{d}{dx} \sec x = \sec x \tan x$.
• The derivative of the tangent function is given by $\frac{d}{dx} \tan x = \sec^2 x$.
• The derivative of the exponential function is given by $\frac{d}{dx} e^x = e^x$.
• The product rule states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.
• The chain rule states that if we have a function of the form $f(x) = u(g(x))$, then the derivative is given by $f^{\prime}(x) = u^{\prime}(g(x)) \cdot g^{\prime}(x)$.

Final Answer

The final answer is: $\boxed{3 \sec x \tan x + 2 e^x (\sec^2 x + \tan x)}$