Find The Derivative Of The Function:${ Y = 3 \tan \left(\frac{1}{4} X\right) + 5 \cot (5x) - 8 \csc X + E^{-3x} }$ { Y^{\prime}(x) = \square \} (Type An Exact Answer.)

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore how to find the derivative of a complex function using the rules of differentiation.

The Function

The given function is:

${ y = 3 \tan \left(\frac{1}{4} x\right) + 5 \cot (5x) - 8 \csc x + e^{-3x} }$

This function involves trigonometric functions, including tangent, cotangent, cosecant, and exponential functions. To find the derivative of this function, we will apply the rules of differentiation for each component.

Step 1: Differentiate the Trigonometric Functions

The derivative of the tangent function is:

${ \frac{d}{dx} \tan u = \sec^2 u \frac{du}{dx} }$

Using this rule, we can differentiate the tangent function in the given function:

${ \frac{d}{dx} 3 \tan \left(\frac{1}{4} x\right) = 3 \sec^2 \left(\frac{1}{4} x\right) \frac{1}{4} }$

Simplifying, we get:

${ \frac{d}{dx} 3 \tan \left(\frac{1}{4} x\right) = \frac{3}{4} \sec^2 \left(\frac{1}{4} x\right) }$

Similarly, the derivative of the cotangent function is:

${ \frac{d}{dx} \cot u = -\csc^2 u \frac{du}{dx} }$

Using this rule, we can differentiate the cotangent function in the given function:

${ \frac{d}{dx} 5 \cot (5x) = -5 \csc^2 (5x) \cdot 5 }$

Simplifying, we get:

${ \frac{d}{dx} 5 \cot (5x) = -25 \csc^2 (5x) }$

Step 2: Differentiate the Cosecant Function

The derivative of the cosecant function is:

${ \frac{d}{dx} \csc u = -\csc u \cot u \frac{du}{dx} }$

Using this rule, we can differentiate the cosecant function in the given function:

${ \frac{d}{dx} -8 \csc x = 8 \csc x \cot x }$

Step 3: Differentiate the Exponential Function

The derivative of the exponential function is:

${ \frac{d}{dx} e^{ax} = ae^{ax} }$

Using this rule, we can differentiate the exponential function in the given function:

${ \frac{d}{dx} e^{-3x} = -3e^{-3x} }$

Combining the Derivatives

Now that we have differentiated each component of the function, we can combine the derivatives to find the derivative of the entire function:

${ y^{\prime}(x) = \frac{3}{4} \sec^2 \left(\frac{1}{4} x\right) - 25 \csc^2 (5x) + 8 \csc x \cot x - 3e^{-3x} }$

Conclusion

In this article, we have explored how to find the derivative of a complex function using the rules of differentiation. We have applied the rules of differentiation for each component of the function, including trigonometric functions and exponential functions. The resulting derivative is a combination of the derivatives of each component.

Final Answer

The derivative of the function is:

${ y^{\prime}(x) = \frac{3}{4} \sec^2 \left(\frac{1}{4} x\right) - 25 \csc^2 (5x) + 8 \csc x \cot x - 3e^{-3x} }$

Introduction

In our previous article, we explored how to find the derivative of a complex function using the rules of differentiation. In this article, we will answer some common questions related to the derivative of a complex function.

Q: What is the derivative of a tangent function?

A: The derivative of a tangent function is:

${ \frac{d}{dx} \tan u = \sec^2 u \frac{du}{dx} }$

This rule can be applied to any tangent function, including the one in the given function:

${ \frac{d}{dx} 3 \tan \left(\frac{1}{4} x\right) = \frac{3}{4} \sec^2 \left(\frac{1}{4} x\right) }$

Q: What is the derivative of a cotangent function?

A: The derivative of a cotangent function is:

${ \frac{d}{dx} \cot u = -\csc^2 u \frac{du}{dx} }$

This rule can be applied to any cotangent function, including the one in the given function:

${ \frac{d}{dx} 5 \cot (5x) = -25 \csc^2 (5x) }$

Q: What is the derivative of a cosecant function?

A: The derivative of a cosecant function is:

${ \frac{d}{dx} \csc u = -\csc u \cot u \frac{du}{dx} }$

This rule can be applied to any cosecant function, including the one in the given function:

${ \frac{d}{dx} -8 \csc x = 8 \csc x \cot x }$

Q: What is the derivative of an exponential function?

A: The derivative of an exponential function is:

${ \frac{d}{dx} e^{ax} = ae^{ax} }$

This rule can be applied to any exponential function, including the one in the given function:

${ \frac{d}{dx} e^{-3x} = -3e^{-3x} }$

Q: How do I combine the derivatives of different functions?

A: To combine the derivatives of different functions, you need to apply the rules of differentiation for each function and then combine the results. In the case of the given function, we combined the derivatives of the tangent, cotangent, cosecant, and exponential functions to get the final derivative:

${ y^{\prime}(x) = \frac{3}{4} \sec^2 \left(\frac{1}{4} x\right) - 25 \csc^2 (5x) + 8 \csc x \cot x - 3e^{-3x} }$

Q: What is the significance of the derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics.

Conclusion

In this article, we have answered some common questions related to the derivative of a complex function. We have explored the rules of differentiation for different types of functions, including tangent, cotangent, cosecant, and exponential functions. We have also discussed how to combine the derivatives of different functions to get the final derivative.

Final Answer

The derivative of the function is:

${ y^{\prime}(x) = \frac{3}{4} \sec^2 \left(\frac{1}{4} x\right) - 25 \csc^2 (5x) + 8 \csc x \cot x - 3e^{-3x} }$

This derivative represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields.