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Why does ln(x) differentiate to 1/x ?

At first glance, this may seem quite complicated. However, it is simple once you make use of exponents. Let y=ln(x) . This can be written as: e y = e ln(x) e to the power of a natural log cancels out, which ...
CE
Answered by Charlie E. • Maths tutor
15281 Views

Differentiate ln(x)/x

To differentiate this, we use the quotient rule as we have 2 functions of x as a fraction. The quotient rule states that: where u is the numerator function and v is the denominator function. Therefore, u=ln ...
CE
Answered by Charlie E. • Maths tutor
12078 Views

Let R denote the region bounded by the curve y=x^3 and the lines x=0 and x=4. Find the volume generated when R is rotated 360 degrees about the x axis.

The area of a circle is given by (pi)r 2 and the area generated by R can be considered as an infinite number of circular areas. Thus, we can write the area generated by R as the integral of (pi)(x 3 ) 2 betw...
SB
Answered by Stephen B. • Maths tutor
5941 Views

Does the equation: x^2+5x-6 have two real roots? If so what are they?

Having two real roots is equivalent to having two intersections with the x axis. The equation is a parabola (a U shape) so, having two real roots is equivalent to having the lowest point of the curve below t...
SB
Answered by Stephen B. • Maths tutor
5798 Views

Differentiate a^x

Set y=a^x2. Take the natural log of both sides: ln(y)=ln(a^x)3. Using the log rules, simplify: ln(y)=xln(a)4. Differentiate both sides with respect to x: 1/y dy/dx=lna+05. Rearrange: dy/dx=yln(a)6. Using the...
HK
Answered by Hafsah K. • Maths tutor
17699 Views