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Given that x=ln(t) and y=4t^3,a) find an expression for dy/dx, b)and the value of t when d2y/dx2 =0.48. Give your answer to 2 decimal place.

a) Firstly, differentiate x and y with respect to t.

Giving you dx/dt = 1/t and dy/dt = 12t²

dy/dx is found using the chain rule:

dy/dx = dy/dt x dt/dx = 12t³

Answered by Sara W. • Maths tutor

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Calculate the gradient of the function y=x^2+6x when y=-9

-9=x^2+6x

0=x^2+6x+9

0=(x+3)(x+3)

when y=-9 x=3

dy/dx=2x+6

dy/dx=2(3)+6=12

Answered by Hassan M. • Maths tutor

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For a curve of gradient dy/dx = (2/(x^2))-x/4, determine a) d^2y/dx^2 b) the stationary point where y=5/2 c) whether this is a maximum or minmum point and d) the equation of the curve

a) Differentiating gives d²y/dx²=-4x^-3-1/4

b) Let dy/dx=0 and rearrange to find x=2

c) Inserting x=2 into d²y/dx²=-4x^-3-1/4 ...

Answered by Katie M. • Maths tutor

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Find ∫ ( 2x^4 - 4x^(-0.5) + 3 ) dx

When integrating, you need to add one to the power and divide the term by the power. We will consider each term individually, 2x⁴ will become (2x⁴⁺¹)/(4+1) = (2x⁵)/5, -4x<...

Answered by Rebecca M. • Maths tutor

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A curve has equation y = f(x) and passes through the point (4,22). Given that f'(x) = 3x^2 - 3x^(1/2) - 7 use intergration to find f(x).

As f '(x) is the differential of f(x) we have to first integrate f '(x). To do this we take each term individually and integrate it. So starting with 3x², to integrate a simple function of x li...

Answered by Jonathan O. • Maths tutor

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