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Question 1
Let $g (x) = csc x + cot x$ , for $0 < x < π$ . Which of the following is the correct expression for $g^{'} (x)$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $- csc x cot x - csc^{2} x$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the functions and their standard derivatives
We need to differentiate $g (x) = csc x + cot x$ term by term. The standard derivatives are $\frac{d}{d x} (csc x) = - csc x cot x$ and $\frac{d}{d x} (cot x) = - csc^{2} x$ .
Step 2: Differentiate each term
Differentiating $csc x$ gives $- csc x cot x$ . Differentiating $cot x$ gives $- csc^{2} x$ . Both co-function derivatives carry a negative sign.
Step 3: Combine the results
Adding the two derivatives: $g^{'} (x) = - csc x cot x + (- csc^{2} x) = - csc x cot x - csc^{2} x$ .
Method #2Approach 2
Step 1: Identify what is being tested
We need the correct signs for both $\frac{d}{d x} (csc x)$ and $\frac{d}{d x} (cot x)$ . The key fact is that all co-function derivatives carry a negative sign.
Step 2: Eliminate options with incorrect signs on both terms
The option $csc x cot x + csc^{2} x$ has both signs positive, which is incorrect since both co-function derivatives are negative. Eliminate this option.
Step 3: Eliminate options with mixed-sign errors
The option $- csc x cot x + csc^{2} x$ has the correct sign on the first term but incorrectly makes $\frac{d}{d x} (cot x) = + csc^{2} x$ . This is wrong. Eliminate it.
Step 4: Eliminate the remaining incorrect option
The option $csc x cot x - csc^{2} x$ has the wrong sign on the first term (should be negative) but correct sign on the second. This is incorrect. Eliminate it.
Step 5: Select the correct answer
The only remaining option is $- csc x cot x - csc^{2} x$ , which correctly applies negative signs to both co-function derivatives.
Question 2
The curve $C$ is defined implicitly by $y^{2} + sin x = 1$ , for $0 \leq x \leq π$ . Which expression gives $\frac{d y}{d x}$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $- \frac{c o s x}{2 y}$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Recognise implicit differentiation is required
The equation $y^{2} + sin x = 1$ cannot be easily solved for $y$ explicitly, so we differentiate both sides with respect to $x$ implicitly.
Step 2: Differentiate both sides with respect to $x$
Differentiating the left side: $\frac{d}{d x} (y^{2}) + \frac{d}{d x} (sin x) = \frac{d}{d x} (1)$ . Using the chain rule on $y^{2}$ : $2 y \frac{d y}{d x} + cos x = 0$ .
Step 3: Solve for $\frac{dy}{dx}$
Rearranging: $2 y \frac{d y}{d x} = - cos x$ , so $\frac{d y}{d x} = - \frac{cos x}{2 y}$ .
Method #2Approach 2
Step 1: Identify the structure of each option
All options involve $cos x$ and $2 y$ . The key questions are: what is the sign, and which goes in numerator versus denominator?
Step 2: Eliminate options with $y$ in the numerator
The options $- \frac{2 y}{c o s x}$ and $\frac{2 y}{c o s x}$ have $y$ in the numerator. Since implicit differentiation of $y^{2}$ gives $2 y \frac{d y}{d x}$ in the equation, solving for $\frac{d y}{d x}$ places $2 y$ in the denominator. Eliminate both.
Step 3: Determine the correct sign
From $2 y \frac{d y}{d x} + cos x = 0$ , moving $cos x$ to the right gives a negative sign: $\frac{d y}{d x} = - \frac{c o s x}{2 y}$ . The option $\frac{c o s x}{2 y}$ (positive) is therefore incorrect. Eliminate it.
Step 4: Select the correct answer
The only remaining option is $- \frac{cos x}{2 y}$ , which is correct.

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Free preview 2/15

Question 1
Let $g (x) = csc x + cot x$ , for $0 < x < π$ . Which of the following is the correct expression for $g^{'} (x)$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $- csc x cot x - csc^{2} x$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the functions and their standard derivatives
We need to differentiate $g (x) = csc x + cot x$ term by term. The standard derivatives are $\frac{d}{d x} (csc x) = - csc x cot x$ and $\frac{d}{d x} (cot x) = - csc^{2} x$ .
Step 2: Differentiate each term
Differentiating $csc x$ gives $- csc x cot x$ . Differentiating $cot x$ gives $- csc^{2} x$ . Both co-function derivatives carry a negative sign.
Step 3: Combine the results
Adding the two derivatives: $g^{'} (x) = - csc x cot x + (- csc^{2} x) = - csc x cot x - csc^{2} x$ .
Method #2Approach 2
Step 1: Identify what is being tested
We need the correct signs for both $\frac{d}{d x} (csc x)$ and $\frac{d}{d x} (cot x)$ . The key fact is that all co-function derivatives carry a negative sign.
Step 2: Eliminate options with incorrect signs on both terms
The option $csc x cot x + csc^{2} x$ has both signs positive, which is incorrect since both co-function derivatives are negative. Eliminate this option.
Step 3: Eliminate options with mixed-sign errors
The option $- csc x cot x + csc^{2} x$ has the correct sign on the first term but incorrectly makes $\frac{d}{d x} (cot x) = + csc^{2} x$ . This is wrong. Eliminate it.
Step 4: Eliminate the remaining incorrect option
The option $csc x cot x - csc^{2} x$ has the wrong sign on the first term (should be negative) but correct sign on the second. This is incorrect. Eliminate it.
Step 5: Select the correct answer
The only remaining option is $- csc x cot x - csc^{2} x$ , which correctly applies negative signs to both co-function derivatives.
Question 2
The curve $C$ is defined implicitly by $y^{2} + sin x = 1$ , for $0 \leq x \leq π$ . Which expression gives $\frac{d y}{d x}$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $- \frac{c o s x}{2 y}$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Recognise implicit differentiation is required
The equation $y^{2} + sin x = 1$ cannot be easily solved for $y$ explicitly, so we differentiate both sides with respect to $x$ implicitly.
Step 2: Differentiate both sides with respect to $x$
Differentiating the left side: $\frac{d}{d x} (y^{2}) + \frac{d}{d x} (sin x) = \frac{d}{d x} (1)$ . Using the chain rule on $y^{2}$ : $2 y \frac{d y}{d x} + cos x = 0$ .
Step 3: Solve for $\frac{dy}{dx}$
Rearranging: $2 y \frac{d y}{d x} = - cos x$ , so $\frac{d y}{d x} = - \frac{cos x}{2 y}$ .
Method #2Approach 2
Step 1: Identify the structure of each option
All options involve $cos x$ and $2 y$ . The key questions are: what is the sign, and which goes in numerator versus denominator?
Step 2: Eliminate options with $y$ in the numerator
The options $- \frac{2 y}{c o s x}$ and $\frac{2 y}{c o s x}$ have $y$ in the numerator. Since implicit differentiation of $y^{2}$ gives $2 y \frac{d y}{d x}$ in the equation, solving for $\frac{d y}{d x}$ places $2 y$ in the denominator. Eliminate both.
Step 3: Determine the correct sign
From $2 y \frac{d y}{d x} + cos x = 0$ , moving $cos x$ to the right gives a negative sign: $\frac{d y}{d x} = - \frac{c o s x}{2 y}$ . The option $\frac{c o s x}{2 y}$ (positive) is therefore incorrect. Eliminate it.
Step 4: Select the correct answer
The only remaining option is $- \frac{cos x}{2 y}$ , which is correct.

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AHL 5.15—Further derivatives and indefinite integration of these, partial fractions

Step-by-step walkthrough

Step 1: Identify the functions and their standard derivatives

Step 2: Differentiate each term

Step 3: Combine the results

Step 1: Identify what is being tested

Step 2: Eliminate options with incorrect signs on both terms

Step 3: Eliminate options with mixed-sign errors

Step 4: Eliminate the remaining incorrect option

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Recognise implicit differentiation is required

Step 2: Differentiate both sides with respect to $x$

Step 3: Solve for $\frac{dy}{dx}$

Step 1: Identify the structure of each option

Step 2: Eliminate options with $y$ in the numerator

Step 3: Determine the correct sign

Step 4: Select the correct answer

13 more questions in this topic

AHL 5.15—Further derivatives and indefinite integration of these, partial fractions

Step-by-step walkthrough

Step 1: Identify the functions and their standard derivatives

Step 2: Differentiate each term

Step 3: Combine the results

Step 1: Identify what is being tested

Step 2: Eliminate options with incorrect signs on both terms

Step 3: Eliminate options with mixed-sign errors

Step 4: Eliminate the remaining incorrect option

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Recognise implicit differentiation is required

Step 2: Differentiate both sides with respect to $x$

Step 3: Solve for $\frac{dy}{dx}$

Step 1: Identify the structure of each option

Step 2: Eliminate options with $y$ in the numerator

Step 3: Determine the correct sign

Step 4: Select the correct answer

13 more questions in this topic