Question

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ g(t)=\sqrt[4]{t}+6 \csc t $$

Short answer

The derivative of \( g(t)=\sqrt[4]{t}+6 \csc t \) is \( g'(t) = (1/4)t^{-3/4} - 6 \csc t \cot t \).

Step-by-step solution

- Transformation of the function into suitable format

Transform the given function into a format in which differentiation will be easier to apply. Rewrite the function as \( g(t) = t^{1/4} - 6 \csc t \).

- Apply differentiation

Differentiate both terms separately by applying the power rule for fractional exponents and the derivative rule for the trigonometric function.Differentiate \( t^{1/4} \) by applying the power rule: \( (1/4)t^{1/4 - 1} = (1/4)t^{-3/4} \). Differentiate \( \csc t \) using standard derivative rule for cosecant to get: \( - \csc t \cot t \). The derivative of the second term becomes: \( -6 \csc t \cot t \).

- Collect derivation results

Combine the derived results from the above step. The derivative of the function will be the sum of the two above results: \( g'(t) = (1/4)t^{-3/4} - 6 \csc t \cot t \).

Key concepts

Power Rule

The power rule is a fundamental concept that helps us find the derivative of functions that are expressed as powers of a variable. If we have a function of the form \( f(x) = x^n \), its derivative is determined by multiplying the exponent by the coefficient and then subtracting one from the exponent.
For example, in the function \( g(t) = t^{1/4} \), we apply the power rule to find the derivative.
Using the power rule, the derivative is \( \frac{1}{4}t^{1/4 - 1} = \frac{1}{4}t^{-3/4} \).

• Multiply the exponent, \( \frac{1}{4} \), by the coefficient (which is 1 in this case).
• Subtract one from the exponent. So, \( 1/4 - 1 = -3/4 \).

This makes the derivative of \( t^{1/4} \) very straightforward to calculate using the power rule.

Cosecant Derivative

The cosecant function, represented as \( \csc(t) \), is one of the basic trigonometric functions and its derivative can be found using trigonometric differentiation rules.
The derivative of \( \csc(t) \) is \(-\csc(t)\cot(t) \).
This means when differentiating expressions involving \( \csc(t) \), you will often end up with a combination of cosecant and cotangent terms in the result.Applying this to the function \( g(t) = 6 \csc(t) \), we multiply the standard derivative of \( \csc(t) \) by the constant 6:

• The derivative becomes \(-6 \csc(t)\cot(t) \).

By memorizing common derivatives like that of \( \csc(t) \), finding derivatives of more complex trigonometric functions becomes manageable.

Fractional Exponents

Fractional exponents are another way to express roots, and they can be easily manipulated using rules of algebra and calculus.
For instance, the fourth root of \( t \), denoted \( \sqrt[4]{t} \), can be rewritten as \( t^{1/4} \).
This transformation is crucial for applying the power rule when differentiating.What are fractional exponents?

• They transform radicals into exponent form, making it simpler to apply rules like the power rule.
• For example, \( \sqrt[4]{t} = t^{1/4} \), meaning the fourth root of \( t \) is equivalent to raising \( t \) to the power of \( 1/4 \).

Understanding fractional exponents helps in solving differentiation problems as it simplifies expressions into forms that are easier to handle analytically.

Trigonometric Differentiation

Trigonometric differentiation involves applying specific rules to find the derivatives of trigonometric functions like \( \sin(t) \), \( \cos(t) \), and \( \csc(t) \).
These functions have unique derivative rules that are essential for calculus.The basic trigonometric derivatives are:

• \( (\sin(t))' = \cos(t) \)
• \( (\cos(t))' = -\sin(t) \)
• \( (\csc(t))' = -\csc(t)\cot(t) \)

Using these rules, you can differentiate more complex expressions that contain trigonometric terms.
For instance, in \( g(t) = 6\csc(t) \), we identify \( \csc(t) \) and use its derivative form \(-\csc(t)\cot(t) \), combining it with coefficient 6 to find the derivative as \(-6\csc(t)\cot(t) \). Understanding and applying these fundamental trigonometric derivatives can greatly simplify calculus problems.