Differentiate each of the following:
Differentiate the following, expressing your answer in factorised form where possible:
Differentiate the following:
Differentiate the following:
Consider the expression e^{\cos x} \sin \left(e^{x}\right).
If u = e^{\cos x}, find \dfrac{d u}{d x}.
If v = \sin \left(e^{x}\right), find \dfrac{d v}{d x}.
Hence, find the derivative of y = e^{\cos x} \sin \left(e^{x}\right).
Consider the equation y = \dfrac{4 x^{2} + e^{x}}{\cos 7 x}.
If u = 4 x^{2} + e^{x}, find u'.
If v = \cos 7x, find v'.
Hence, find y'.
Find the gradient of f \left( x \right) = \left(x - 8\right)^{3} \left(x - 4\right)^{4} at x = 6.
Find the gradient of f \left( x \right) = \dfrac{\left(x + 7\right)^{9}}{\left(x + 9\right)^{4}} at x = - 5.
Consider the function h \left( t \right) = \left(3 + t\right) \left(3 - t^{3}\right)^{4}.
Determine the expression for h' \left( t \right).
Hence, calculate the gradient of the function h \left( t \right) where t = 1.
Consider the function h \left( t \right) = \left(1 - 4 t^{5}\right)^{4}.
Determine the expression for h' \left( t \right).
Hence, find the exact values of t at which gradient of h \left( t \right) is zero.
Consider the function y = \left( 3 x + 2\right) \sqrt{3 + 8 x}.
Differentiate the function.
Hence, determine the value of x at which the gradient of the function is zero.
Consider the function f\left(x\right)=e^{5x}\left(e^x+e^{-x}\right).
Find f' \left( x \right).
Find f' \left( 0 \right).
Explain why the gradient of the function y = \sin ^{2}\left( 5 x\right) + \cos ^{2}\left( 5 x\right) is equal to 0 for all x.
If f \left( x \right) = \dfrac{\cos ^{2}\left(x\right)}{1 + \sin x}, evaluate f' \left( \dfrac{\pi}{4} \right).
If f \left( t \right) = \sqrt{1 + \sin ^{2}\left(t\right)}, evaluate f' \left( \dfrac{\pi}{2} \right).
For each of the following curves and given points:
Find an expression for \dfrac{dy}{dx}.
Find the exact value of the gradient of the curve at the given point.
y = \sin ^{2}\left( 4 x\right) at x = \dfrac{\pi}{32}.
y = \cos ^{2}\left( 2 x\right) at x = \dfrac{\pi}{24}.
Consider the curve y = \left(x - 8\right)^{2} \left( 2 x + 5\right). Find the equation of the tangents to the curve at the x-intercepts of the graph.
Find the equation of the tangent to the following curves:
y = e^{x} - 3 \sin x at x = \dfrac{3 \pi}{2}.
y = e^{\cos x} at x = \dfrac{3 \pi}{2}.
y = x \cos x at \left(\dfrac{\pi}{2}, 0\right).
y = - 5 x \sin x at \left(\pi, 0\right).