Photo AI

Given that $y = 5x^3 + 7x + 3$, find (a) $ rac{dy}{dx}$ - Edexcel - A-Level Maths Pure - Question 4 - 2005 - Paper 2

Question icon

Question 4

Given-that-$y-=-5x^3-+-7x-+-3$,-find--(a)-$-rac{dy}{dx}$-Edexcel-A-Level Maths Pure-Question 4-2005-Paper 2.png

Given that $y = 5x^3 + 7x + 3$, find (a) $ rac{dy}{dx}$. (b) $ rac{d^2y}{dx^2}$. (ii) Find $igg[1 + 3 rac{ ext{√}}{x} - rac{1}{x^2}igg] dx$.

Worked Solution & Example Answer:Given that $y = 5x^3 + 7x + 3$, find (a) $ rac{dy}{dx}$ - Edexcel - A-Level Maths Pure - Question 4 - 2005 - Paper 2

Step 1

(a) $ rac{dy}{dx}$

96%

114 rated

Answer

To find the first derivative of the function y=5x3+7x+3y = 5x^3 + 7x + 3, we will apply the power rule of differentiation, which states that if y=axny = ax^n, then rac{dy}{dx} = nax^{n-1}.

Calculating: rac{dy}{dx} = 15x^2 + 7

Step 2

(b) $ rac{d^2y}{dx^2}$

99%

104 rated

Answer

To find the second derivative, we differentiate the first derivative we just found:

From rac{dy}{dx} = 15x^2 + 7, apply the power rule again: rac{d^2y}{dx^2} = rac{d}{dx}(15x^2 + 7) = 30x

Step 3

Find $igg[1 + 3 rac{ ext{√}}{x} - rac{1}{x^2}igg] dx$

96%

101 rated

Answer

To find this integral, we will first simplify the expression: 1 + 3 rac{√x}{x} - rac{1}{x^2} = 1 + 3x^{-1/2} - x^{-2}

Now we can integrate each term separately: ∫(1+3x−1/2−x−2) dx\int(1 + 3x^{-1/2} - x^{-2}) \, dx

The integrals are:

  1. ∫1 dx=x\int 1 \, dx = x
  2. ∫3x−1/2 dx=6x1/2\int 3x^{-1/2} \, dx = 6x^{1/2}
  3. ∫−x−2 dx=x−1\int -x^{-2} \, dx = x^{-1}

Thus, combining all parts, we have: x + 6\sqrt{x} + rac{1}{x} + C where CC is the constant of integration.

Join the A-Level students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;