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5. (a) Differentiate $$ rac{ ext{cos} \, 2x}{ ext{√}x}$$ with respect to x - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 8
Question 7
5. (a) Differentiate $$ rac{ ext{cos} \, 2x}{ ext{√}x}$$ with respect to x. (b) Show that $$\frac{d}{dx}(\sec^2 \, 3x)$$ can be written in the form $$\mu(\tan ... show full transcript
Worked Solution & Example Answer:5. (a) Differentiate $$ rac{ ext{cos} \, 2x}{ ext{√}x}$$ with respect to x - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 8
Step 1
Differentiate \(\frac{\text{cos} \, 2x}{\text{√}x}\) with respect to x.
Answer
To differentiate (\frac{\text{cos} , 2x}{\text{√}x}), use the quotient rule:
where (u = \text{cos} , 2x) and (v = \text{√}x).
-
Differentiate (u):
- (u' = -2 \text{sin} , 2x)
-
Differentiate (v):
- (v' = \frac{1}{2\sqrt{x}})
-
Substitute:
- (d/dx\left(\frac{\text{cos} , 2x}{\sqrt{x}}\right) = \frac{(-2 \text{sin} , 2x)\sqrt{x} - \text{cos} , 2x \frac{1}{2\sqrt{x}}}{x})
-
Simplify the expression.
Step 2
Show that \(\frac{d}{dx}(\sec^2 \, 3x)\) can be written in the form \(\mu(\tan \, 3x + \tan^3 \, 3x)\).
Answer
Using the chain rule:
-
Differentiate (\sec^2 , 3x):
- (\frac{d}{dx}(\sec^2 , 3x) = 2 \sec^2(3x) \cdot \sec(3x) \tan(3x) \cdot 3 = 6 \sec^2(3x) \tan(3x))
-
Notice that:
- (\sec^2 3x = 1 + \tan^2 3x)
- Substitute this into the derivative to show:
- (6(1 + \tan^2 3x) \tan 3x = 6 \tan 3x + 6 \tan^3 3x)
-
This can be expressed as (\mu(\tan 3x + \tan^3 3x)) with (\mu = 6).
Step 3
Given \(x = 2 \sin(\frac{y}{3})\), find \(\frac{dy}{dx}\) in terms of \(x\).
Answer
To find (\frac{dy}{dx}):
-
Differentiate the equation:
- (\frac{dx}{dy} = 2 \cdot \frac{1}{3} \cos(\frac{y}{3}))
- Thus, (\frac{dx}{dy} = \frac{2}{3} \cos(\frac{y}{3}))
-
Invert to find (\frac{dy}{dx}):
- (\frac{dy}{dx} = \frac{3}{2 \cos(\frac{y}{3})})
-
Replace (\cos(\frac{y}{3})) using (x):
- Solve for (y) in terms of (x): (y = 3\arcsin(\frac{x}{2})) and substitute to get (\frac{dy}{dx}) in terms of (x).
Therefore, the answer will simplify to:
- (\frac{dy}{dx} = \frac{3}{2\sqrt{1 - (\frac{x}{2})^2/4}}).