Differentiating a function of a function question?

Question

I'm so stuck please help me!
Even if you can just do one question it will help!
Differentiate the following:
1) ln(1/x)
2) sin4x^3
3) cos^2(x^3)
4) tan^2(xe^x)
5) ln(sin^3(x^4))

6) If x and y lie on the ellipse x^2/a^2 + y^2/b^2 = 1, show that the slope at any point is given by dy/dx = -(b/a)^2(x/y)

Thank you if anyone helps me! :) x

Answer 1

you need to know the Chain rule of differentiation. IT is very simple:

let y = f(u) and u = f(x)

then dy/dx = f'(u)*u'(x)

Let us take your first one in this model: ln(1/x)

here y = ln(u) and u = f(x) = 1/x

Therefore dy/du = ln(u)' = 1/u

du/dx = (1/x)' = -1/x**2

therefore dy/dx = (1/(1/x))(-1/x**2) = x(-1/x**2) = -1/x

now try the second one using my model and send me your answer and I will check it out for you.

Answer 2

In each case remember that if f(x) is a composite function of g(x) and h(x) then
f(x) = g(h(x)) => f'(x) = g'(h(x))*h'(x) So the 1'st one is
d/dx ln(1/x) = 1/(1/x) * d/dx (1/x) = x * (-1/x²) = -1/x
The 2'nd one is
d/dx sin(4x^3) = cos(4x^3) d/dx 4x^3 = cos(4x^3)*12x² or
12x²cos(4x^3)
Now you do the rest so you learn it ☺

Doug

Answer 3

1) x
2) cos(4x^3) X 12x^2
Sorry no time g2g

Answer 4

Please review your class notes on the chain rule and on implicit differentiation.