Differentiate each of the following functions:
y = - 4 e^{x}
y = \dfrac{1}{2} e^{ - x }
y = 7 e^{ - 3 x}
y = 6 e^{\frac{x}{3}}
y = e^{ 8 x}
y = e^{ - 2 x } + 4
y = 3 e^{ 5 x} - e^{ - 4 x } + x^{2}
y = e^{ 3 x} + 5 x^{2} - 2
y = 2 e^{ 4 x} - 3 e^{ - 5 x }
y = e^{ 2 x} + e^{9} + e^{ - 5 x }
y = e^{ 5 x} + e^{ 2 x}
y = \dfrac{e^{x} - e^{ 3 x} + 1}{e^{x}}
y = \dfrac{e^{x} - e^{ - x }}{2}
y= e^{ 8 x} - e^{ - 3 x }
y = \dfrac{e^{ 9 x}}{9} + \dfrac{e^{ 8 x}}{8}
y = e^{x} + e^{ - x }
Consider the function y = e^{ a x}, where a is a constant.
Let u = a x. Rewrite the function y in terms of u.
Determine \dfrac{du}{dx}.
Hence determine \dfrac{dy}{dx}.
Consider the function f \left( x \right) = e^{ 2 x}.
Find f' \left( x \right).
Find the exact form of f' \left( - 3 \right).
Consider the function f \left( x \right) = 3 - e^{ - x }.
Determine f \left( 0 \right).
Determine f' \left( 0 \right).
Which of the following statements is true?
f' \left( x \right) < 0 for x \geq 0
f' \left( x \right) < 0 for all real x.
f' \left( x \right) > 0 for all real x.
f \left( x \right) > 0 for all real x.
Determine the value of \lim_{x \to \infty} f \left( x \right).
Consider the function y = e^{ 2 x}.
Find \dfrac{d y}{d x}.
Find the exact value of the derivative when x = 0.
Find the exact value of the derivative when x = 5.
Consider the following functions.
Rewrite the function as a power of e.
Find \dfrac{d y}{d x}.
y = \sqrt{e^{x}}
y = \sqrt[7]{e^{x}}
y = \dfrac{1}{\sqrt{e^{x}}}
Consider the function y = e^{ - x }.
Determine \dfrac{dy}{dx}.
Find the gradient of the tangent to the curve at x = - 2. Round your answer to three decimal places.
For each of the following functions:
For each of the following functions:
Find the equation of the tangent to the curve f \left( x \right) = 2 e^{x} at the point where it crosses the y-axis. Express the equation in the form y = m x + c.
The tangent to the curve f \left( x \right) = 2.5 e^{x} at the point x = 7.5 is parallel to the tangent to the curve g \left( x \right) = e^{ 2.5 x} at the point where x = k. Solve for k.
Point R lies on the curve with equation y = e^{ 3 x + 2}.
If the x-coordinate of point R is - \dfrac{2}{3}, determine its y-coordinate.
Determine the gradient m of the tangent at the point R.
Find the gradient of the normal at point R.
Hence find the equation of the normal at the point R. Write your answer in general form.
Point Q \left( - 1 , e\right) lies on the curve with equation y = e^{ - x }.
Determine the gradient m of the tangent at the point Q.
Find the gradient of the normal at point Q.
Hence find the equation of the normal at the point Q.
State the coordinates of the y-intercept of the normal.
Determine the coordinates of the x-intercept of the normal.
Find the exact area in expanded form of the triangle which has vertices at these intercepts and the origin.
Consider the function y = 7^{x}.
Rewrite the function in terms of natural base e.
Hence determine y'. Express the derivative in terms of the base 7.
Hence determine the exact gradient of the tangent of the curve at x = 1.