Differentiate the following functions:
Consider the function y = \ln \left(\dfrac{x - 3}{x + 3}\right).
Let u = \dfrac{x - 3}{x + 3}. Rewrite y in terms of u.
Find \dfrac{d u}{d x}.
Find \dfrac{d y}{d u} in terms of x.
Find \dfrac{d y}{d x}.
Consider the function y = \ln x^{2}, where x \gt 0.
Rewrite the function without powers.
Hence determine y \rq \left( x \right).
Find the value of x at which y \rq \left( x \right) = \dfrac{1}{4}.
Consider the function y = \ln \left( 5 x - 2\right)^{4}.
Rewrite the function without powers.
Hence determine y \rq \left( x \right).
Find the exact value of x at which y' = \dfrac{1}{3}.
Differentiate the following functions:
Consider the function f \left( x \right) = \ln \left(\sqrt{x^{2} + 1}\right).
Find f \rq \left( x \right).
Find f \rq \left( 2 \right).
Consider the function f \left( x \right) = 4 \ln \left( 4 x^{2} + 3\right).
Find f \rq \left( x \right).
Find x, such that f \rq \left( x \right) = 4.
Differentiate the following functions:
Consider the curve y = x^{3} \ln x.
Find the gradient function \dfrac{d y}{d x}.
Find the exact value of the gradient at the point where x = e^{4}.
If f \left( 6 \right) = 2 and f \rq \left( 6 \right) = 8, find the value of \dfrac{d}{dx} \left(\ln \left(f \left( x \right)\right)\right) at x = 6.
Given that f \left( x \right) = \ln \left(g \left( x \right)\right), g \left( 2 \right) = 4 and g \rq \left( 2 \right) = 9, evaluate f \rq \left( 2 \right).
Consider the function f \left( x \right) = x e^{ 3 x}.
Show that x e^{ 3 x} = e^{ 3 x + \ln x}.
Hence, find f \rq \left( x \right), without using the product rule.
Suppose that g \left( x \right) = \dfrac{\ln x}{f \left( x \right)}, for some function f \left( x \right).
Find an expression for g' \left( x \right) in terms of f \left( x \right) and its derivative f \rq \left( x \right).
If f \left( e^{3} \right) = 4 e^{3} and f \rq \left( e^{3} \right) = 2, find the value of g \rq \left( x \right) when x = e^{3}.
Consider the functions f \left( x \right) = k \ln x and g \left( x \right) = \ln k x, where k \gt 1 is a constant.
Find f \rq \left( x \right).
Find g \rq \left( x \right).
State how many times faster f \left( x \right) is increasing compared to g \left( x \right).
Given the following expressions:
- \ln x = x - 1 - \dfrac{\left(x - 1\right)^{2}}{2} + \dfrac{\left(x - 1\right)^{3}}{3} - \dfrac{\left(x - 1\right)^{4}}{4} + ...
- \dfrac{1}{x} = 1 - \left(x - 1\right) + \left(x - 1\right)^{2} - \left(x - 1\right)^{3} + \left(x - 1\right)^{4} - ...
Prove that \dfrac{d}{dx} \left(\ln x\right) = \dfrac{1}{x}.
Consider the function y = \log_3 x.
Make x the subject.
Rewrite the expression for x with base e.
Find \dfrac{dx}{dy}.
Write \dfrac{dx}{dy} in terms of x.
Hence find \dfrac{dy}{dx}.
Differentiate the following functions: