Consider the graph of y = \ln x.
State whether the function is increasing or decreasing.
State whether the gradient to the curve is negative at any point on the curve.
Describe the change in the gradient of the tangent as x increases.
Describe the change in the gradient of the tangent as x gets closer and closer to 0.
Consider the function f \left( x \right) = \ln x.
Find f\rq \left(5\right).
State the value of x for which f\rq \left(x \right) is undefined.
Consider the function f \left( x \right) = \ln x.
Find the x-intercept.
State the equation of the tangent line to the curve at the point where it crosses the x-axis.
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 = \ln a x, where a is a constant with a\gt 0 and x \gt 0.
Let u = a x. Rewrite the y in terms of u.
Find \dfrac{d u}{d x}.
Hence find \dfrac{d y}{d x}.
Given f \left( x \right) = \ln 6 x. Find:
Differentiate the following functions:
y = 7 \ln x
y = -2\ln x
y = \ln 6 x
y = \ln 4 x - \ln 3
y = \ln \left(\dfrac{x}{7}\right)
y = 2 \ln 3 x
y = 4 \ln \left(\dfrac{x}{5}\right)
y = \ln \left(8 - x\right) - 8
y = \ln \left(2 + 4 x + x^{3}\right)
y = \ln \left( 4 x + 5\right)
y = \ln \left(3 - 2 x\right)
y = 2 \ln \left( 6 x - 5\right)
y = 3 \ln \left( \dfrac{1}{4} x - 6\right)
y = \ln \left( 7 x + 3\right)
y = \ln \left(\sin x\right)
Consider the functions f \left( x \right) = \ln x and g \left( x \right) = \ln 3 x.
Sketch both functions on the same Cartesian plane.
Find f \rq \left( x \right).
Find g \rq \left( x \right).
What can be concluded about the tangents of the curves at any given x-value?
Consider the function y = \ln \left( - x \right).
State the domain of this function.
Find \dfrac{dy}{dx}.
Hence, find the gradient of the tangent to the curve at x = - 5.
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 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).
How many times faster is f \left( x \right) increasing than g \left( x \right)?
For each of the following curves:
Find the derivative of the function.
Find the exact value of the gradient of the tangent at the given point.
y = \ln 2 x at the point where x=5.
y = 4 x + \ln 3 x at the point where x = \dfrac{1}{5}.
y = \ln \left(x^{2} + 5\right) at the point where x=3.
y = \ln \left(x^{4} + 4\right) at the point where x=2.
y = \ln \left( \sqrt x\right) at the point where x=e^2.
y = \ln \left(x - 3\right) at the point where x=4.
y = \ln \left( 3 x - 2\right) at the point where x=1.
y = x^{2} \ln x^{2} at the point where x = e.
At the point \left(a, b\right) on the curve y = \ln \left( - 2 x\right), the gradient of the tangent to the curve is - \dfrac{1}{3}. Determine the value of a.
For each of the following functions:
State the domain of y.
Find y \rq.
State the domain of y \rq.
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.
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).
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.
Differentiate the following functions:
y = 5\ln \sqrt{x}
y = \ln \left(\sqrt{ 7 x}\right)
y = \ln \left(\sqrt{2 - 5 x}\right)
y = \dfrac{\ln x}{x}
y = \dfrac{x}{\ln x}
y = \ln \left(\dfrac{8}{x + 4}\right)
y = \ln \left(\dfrac{4}{3 x + 4}\right)
y = \dfrac{\ln 3 x}{5 x^{4} + 2}
f \left( x \right) = \ln \left(\left( 4 x^{3} + 8 x^{2} - 9\right)^{3}\right)
y = 6 x^{4} \ln x
y = x \ln x
y = \left(\ln x\right)^{6}
y = 3 \ln x + 5 \ln 2 x
y = x^{4} + 6 \ln 7 x
y = 4 \ln x - \dfrac{1}{x}
y = \ln \left(\dfrac{3}{x}\right)
y = \left(x + 6\right) \ln \left(x + 6\right)
y = x^{5} \ln \left(x - 5\right)
y = \left(x^{2} + 6\right) \ln 2 x
Differentiate the following functions:
y = \dfrac{\ln x}{\sin x}
y = \cos \left(\ln x\right)
y = \cos x \ln x
y = \ln e^{ 8 x}
y = e^{\ln x}
y = \log_{2} x
y = e^{x} \ln \left(x\right)
y = \dfrac{\ln x}{e^{x}}
y = \dfrac{\ln x}{e^{ 2 x}}
y = e^{ - 2 } \ln \left( - 6 + x^{ - 3 }\right)
y = \ln \left(\log_{e} 4 x\right)
y = \ln \left(\ln x\right)
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}.
Consider the function y = x e^{x}.
Show that e^{x + \ln x} = x e^{x}.
Hence, find \dfrac{d y}{d x}, without using the product rule.
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.
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).
Determine \dfrac{d y}{d x}, given that y = u^{5} and u = \ln \left(x + 5\right).
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}.
Consider the function y = \ln \left(\ln x \right).
Find the derivative of y.
Find y when x = e.
Hence, find the equation of the tangent to the curve at x = e.
Consider the function y = 2 \ln \left(x^{2} + e \right).
Find the derivative of y.
Evaluate the derivative at x = 0.
Hence, find the equation of the tangent to the curve at x = 0.
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 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 \left(\dfrac{1}{\left(x - 4\right)^{3}}\right).
Simplify the function.
Find \dfrac{d y}{d x}.
State the values of x for which the function is undefined.
Determine whether the function is strictly increasing, strictly decreasing, or neither. Explain your answer.
Consider the function f \left( x \right) = \ln x^{4}.
Find f \rq \left( x \right).
Find f \rq \left( 2 \right).
Find f \rq \rq \left( x \right).
Find f \rq \rq \left( 2 \right).
State whether the function is increasing or decreasing at x = 2.
Describe the concavity of the function at x = 2.
Consider the function f \left( x \right) = \dfrac{\ln x}{x} for x \gt 0.
Find the x-coordinate of the stationary point.
State whether the stationary point is a maximum or minimum.
Consider the function f \left( x \right) = x^{2} \ln x.
State the domain of this function.
Find the exact coordinates of the turning point.
Determine the nature of the turning point.
Consider the function y = x \ln x.
State the domain of the function.
Find y \rq.
Determine the exact coordinates of the turning point.
Find y \rq \rq at the turning point.
State whether the turning point is a maximum or minimum value of the function.
State the range of the function in exact form.
Describe the behaviour of the function as x \to \infty.
Sketch the graph of the function.