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Explain why for any constant a, if y = a^x then dy/dx = a^x(ln(a))

So let's start with taking the natural log on both sides of y=a^x, giving us ln(y) = ln(a^x). Using the laws of logarithms we can write this as ln(y) = xln(a).Next, we differentiate bo...

Answered by James M. • Maths tutor

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Find the coordinates and determine the nature of the stationary points of curve y=(2/3)x^3+2x^2-6x+3

Stationary points occur when dy/dx=0, therefore determine dy/dx first:
dy/dx= 2x²+ 4x - 6
2) solve dy/dx=0 for two values of x (using quadratic formula, if necessary):
(x+...

Answered by Barbora V. • Maths tutor

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Solve for x and y: 2x +5y + 5= 0 , 2y + 31= 5x

For a question like this you should aim to eliminate either x OR y from one equation in order to deduce the value of the other. 1) 2x +5y + 5= 0 , 2) 2y + 31= 5x
Rearrange equation 2) so that 2y +31...

Answered by Tutor114325 D. • Maths tutor

4877 Views

The normal to the curve x*(e^-y) + e^y = 1 + x, at the point (c,lnc), has a y-intercept c^2 + 1. Determine the value of c.

We firstly need to find the slope of the line normal to the curve at the point (c, lnc). To do this, we will differentiate implicitly the equation of the curve to find the slope of the tangent. By implici...

Answered by Katerina A. • Maths tutor

6272 Views

A cuboid has length x cm. The width of the cuboid is 4 cm less than its length. The height of the cuboid is half of its length. The surface area of the cuboid is 90 cm^2 . Show that 2x^2 − 6x − 45 = 0

Take each side of the cuboid as an algebraic expression and multiply each by 2 to account for both sides of the shape. For example, (x)(x-4), which could be expanded to x²-4x, and then multipl...

Answered by Tom M. • Maths tutor

10380 Views