Mastering Derivatives: Calculus Study Guide

Q1. What is the derivative of $\csc^{-1}(u)$ with respect to $x$?

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your knowledge of how to differentiate the inverse cosecant function, where $u$ is a differentiable function of $x$.

Key Terms and Formulas

• $\csc^{-1}(u)$: Inverse cosecant function

• Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Step-by-Step Guidance

1. Recall the general formula for the derivative of $\csc^{-1}(u)$.

2. Apply the chain rule since $u$ is a function of $x$.

3. Set up the derivative expression, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer:

$\left(\csc^{-1}\left(u\right)\right)^{\prime}=-\frac{1}{\left\vert u\right\vert\sqrt{u^2-1}}\cdot u^{\prime}$

This formula uses the chain rule and the standard derivative for the inverse cosecant function.

Q2. What is the derivative of $\cos(u)$ with respect to $x$?

Background

Topic: Derivatives of Trigonometric Functions

This question tests your ability to differentiate the cosine function when its argument is itself a function of $x$.

Key Terms and Formulas

• $\cos(u)$: Cosine function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\cos(u)$ with respect to $u$.

2. Apply the chain rule to account for $u$ being a function of $x$.

3. Set up the derivative expression, but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\cos(u) = -\sin(u) \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite function.

Q3. What is the derivative of $\csc(u)$ with respect to $x$?

Background

Topic: Derivatives of Trigonometric Functions

This question tests your ability to differentiate the cosecant function when its argument is a function of $x$.

Key Terms and Formulas

• $\csc(u)$: Cosecant function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\csc(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

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

The chain rule is used to differentiate the composite function.

Q4. What is the derivative of $\sin(u)$ with respect to $x$?

Background

Topic: Derivatives of Trigonometric Functions

This question tests your ability to differentiate the sine function when its argument is a function of $x$.

Key Terms and Formulas

• $\sin(u)$: Sine function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\sin(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\sin(u) = \cos(u) \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite function.

Q5. What is the derivative of $\sec(u)$ with respect to $x$?

Background

Topic: Derivatives of Trigonometric Functions

This question tests your ability to differentiate the secant function when its argument is a function of $x$.

Key Terms and Formulas

• $\sec(u)$: Secant function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\sec(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\sec(u) = \sec(u)\tan(u) \cdot \frac{du}{dx}$

The chain rule is used to differentiate the composite function.

Q6. What is the derivative of $\cot(u)$ with respect to $x$?

Background

Topic: Derivatives of Trigonometric Functions

This question tests your ability to differentiate the cotangent function when its argument is a function of $x$.

Key Terms and Formulas

• $\cot(u)$: Cotangent function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\cot(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

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

The chain rule is applied to differentiate the composite function.

Q7. What is the derivative of $\log_a(u)$ with respect to $x$?

Background

Topic: Derivatives of Logarithmic Functions

This question tests your ability to differentiate logarithmic functions with a base $a$.

Key Terms and Formulas

• $\log_a(u)$: Logarithm base $a$

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\log_a(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\log_a(u) = \frac{1}{u \ln a} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite logarithmic function.

Q8. What is the derivative of $e^u$ with respect to $x$?

Background

Topic: Derivatives of Exponential Functions

This question tests your ability to differentiate the exponential function with base $e$.

Key Terms and Formulas

• $e^u$: Exponential function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $e^u$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\left(e^{u}\right)^{\prime}=e^{u}\cdot u^{\prime}$

The chain rule is applied to differentiate the composite exponential function.

Q9. What is the derivative of $\tan^{-1}(u)$ with respect to $x$?

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your ability to differentiate the inverse tangent function.

Key Terms and Formulas

• $\tan^{-1}(u)$: Inverse tangent function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\tan^{-1}(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\tan^{-1}(u) = \frac{1}{1+u^2} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite inverse tangent function.

Q10. What is the derivative of $\cot^{-1}(u)$ with respect to $x$?

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your ability to differentiate the inverse cotangent function.

Key Terms and Formulas

• $\cot^{-1}(u)$: Inverse cotangent function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\cot^{-1}(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\cot^{-1}(u) = -\frac{1}{1+u^2} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite inverse cotangent function.

Q11. What is the derivative of $\cos^{-1}(u)$ with respect to $x$?

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your ability to differentiate the inverse cosine function.

Key Terms and Formulas

• $\cos^{-1}(u)$: Inverse cosine function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\cos^{-1}(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\cos^{-1}(u) = -\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite inverse cosine function.

Q12. What is the derivative of $\sec^{-1}(u)$ with respect to $x$?

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your ability to differentiate the inverse secant function.

Key Terms and Formulas

• $\sec^{-1}(u)$: Inverse secant function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\sec^{-1}(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\sec^{-1}(u) = \frac{1}{|u|\sqrt{u^2-1}} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite inverse secant function.

Q13. What is the derivative of $a^u$ with respect to $x$?

Background

Topic: Derivatives of Exponential Functions

This question tests your ability to differentiate exponential functions with base $a$.

Key Terms and Formulas

• $a^u$: Exponential function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $a^u$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}a^u = a^u \ln a \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite exponential function.

Q14. What is the derivative of $\tan(u)$ with respect to $x$?

Background

Topic: Derivatives of Trigonometric Functions

This question tests your ability to differentiate the tangent function when its argument is a function of $x$.

Key Terms and Formulas

• $\tan(u)$: Tangent function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\tan(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

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

The chain rule is applied to differentiate the composite tangent function.

Q15. What is the derivative of $\ln(u)$ with respect to $x$?

Background

Topic: Derivatives of Logarithmic Functions

This question tests your ability to differentiate the natural logarithm function.

Key Terms and Formulas

• $\ln(u)$: Natural logarithm function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\ln(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\ln(u) = \frac{1}{u} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite logarithmic function.

Q16. What is the derivative of $\sin^{-1}(u)$ with respect to $x$?

Background

Topic: Derivatives of Inverse Trigonometric Functions

This question tests your ability to differentiate the inverse sine function.

Key Terms and Formulas

• $\sin^{-1}(u)$: Inverse sine function

• Chain Rule

Step-by-Step Guidance

1. Recall the derivative of $\sin^{-1}(u)$ with respect to $u$.

2. Apply the chain rule for $u$ as a function of $x$.

3. Set up the derivative expression, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\sin^{-1}(u) = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$

The chain rule is applied to differentiate the composite inverse sine function.

Q17. Find $\frac{d}{dx}\left(\frac{u}{v}\right)$ where $u$ and $v$ are differentiable functions of $x$.

Background

Topic: Quotient Rule

This question tests your ability to use the quotient rule to differentiate a ratio of two functions.

Key Terms and Formulas

• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

Step-by-Step Guidance

1. Identify $u$ and $v$ as differentiable functions of $x$.

2. Recall the quotient rule formula.

3. Set up the expression for the derivative, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

This formula is the standard quotient rule for differentiation.

Q18. Find $(uv)'$ where $u$ and $v$ are differentiable functions of $x$.

Background

Topic: Product Rule

This question tests your ability to use the product rule to differentiate the product of two functions.

Key Terms and Formulas

• Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$

Step-by-Step Guidance

1. Identify $u$ and $v$ as differentiable functions of $x$.

2. Recall the product rule formula.

3. Set up the expression for the derivative, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$

This formula is the standard product rule for differentiation.

Q19. Find $\frac{dy}{dx}$ where $y = f^{-1}(x)$.

Background

Topic: Derivative of Inverse Functions

This question tests your ability to differentiate an inverse function.

Key Terms and Formulas

• Inverse Function: $y = f^{-1}(x)$

• Derivative Formula: $\frac{dy}{dx} = \frac{1}{f'(y)}$

• solve for y': $y^{\prime}=\frac{1}{f^{\prime}\left(f^{-1}\left(x\right)\right)}$

Step-by-Step Guidance

1. Recall the formula for the derivative of an inverse function.

2. Set up the expression for the derivative, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$y^{\prime}=\frac{1}{f^{\prime}\left(f^{-1}\left(x\right)\right)}$

This formula gives the derivative of the inverse function at $x$.

Q20. Find the derivative of $f(g(x))$ with respect to $x$.

Background

Topic: Chain Rule

This question tests your ability to differentiate a composite function using the chain rule.

Key Terms and Formulas

• Composite Function: $f(g(x))$

• Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Step-by-Step Guidance

1. Identify the outer function $f$ and the inner function $g$.

2. Recall the chain rule formula.

3. Set up the expression for the derivative, but do not compute the final value.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

This formula is the standard chain rule for differentiation.