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Given that y=(4x^2)lnx, find f"(x) when x=e^2

Product rule: first order derivative = 8xlnx+4x Second order derivative = 8lnx+12 Sub in e^2 , 16lne + 12 = 24
JH
Answered by Johnny H. • Maths tutor
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What is the difference between differentiation and integration, and why do we need Calculus at all?

We need calculus to find out the properties complex curves that we cannot find with simple arithmetic as we would with linear graphs. Differentiation allows us to calculate the gradient of a curve at a speci...
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Answered by Samuel H. • Maths tutor
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The equation (t – 1)x^2 + 4x + (t – 5) = 0, where t is a constant has no real roots. Show that t satisfies t2–6t+1>0

This is a quadratic equation of the form ax^2 + bx +c, where (t-1)=a 4=b . (t-5)=0 Therefore if there are no real roots, you get that b^2-4ac<0. (using the quadratic formula) First plug in values for a b ...
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Answered by Rachel Y. • Maths tutor
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Question 3 on the OCR MEI C3 June 2015 paper. Find the exact value of Integral x^3 ln x dx between 1 and 2.

This is a good candidate for integration by parts because it is a product of two functions – ln(x) and x^3, and x^3 is easy to integrate. The definition for integration by parts is: integral(u dv/dx) dx = uv...
TK
Answered by Tom K. • Maths tutor
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Question 3 on the OCR MEI C1 June 2015 paper. Evaluate the following. (i) 200^0 (ii) (9/25)^(-1/2)

For both parts you have to remember some of the properties of powers. Rule 1: x^0 = 1. Rule 2: x^(-n) = 1/(x^n). Rule 3: x^(1/n) = n-th root of x. For part (i) we can use rule 1 to see that 200^0 = 0. For pa...
TK
Answered by Tom K. • Maths tutor
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