Differentiate the following:
Find the derivative of y = x^{6} + x^{\frac{1}{5}}. Write your answer with positive indices.
Differentiate y = 18 x \sqrt{x}. Leave your answer in surd form.
Let y = \left( 4 x + 3\right)^{ - 1 } be defined as a composition of the functions y = u^{ - 1 } and u = 4 x + 3. Find:
\dfrac{d y}{d u}
\dfrac{d u}{d x}
\dfrac{dy}{dx}
Consider the function y = \left( 2 x^{4} + 6\right)^{5} where u = 2 x^{4} + 6.
Find \dfrac{d u}{d x}.
Express y as a function of u.
Find \dfrac{d y}{d u}.
Hence find \dfrac{d y}{d x}.
Consider the function f \left( x \right) = \left( 5 x^{3} - 4 x^{2} + 3 x - 5\right)^{7}.
Redefine the function as composite functions f \left( u \right) and u \left( x \right), where u \left( x \right) is a polynomial.
Differentiate f(x) using the chain rule.
Differentiate each of the following:
Differentiate y = 4 x^{5} + \dfrac{1}{x - 4} + 6. Express the answer using positive indices.
Consider the function y = x^{6} \left(x^{4} + 4\right). Let u = x^{6} and v = x^{4} + 4. Find:
u'
v'
\dfrac{dy}{dx}
Differentiate the following using the product rule:
Differentiate y = \left( 3 x + 2\right) \sqrt{5 + 4 x}. Express your answer as a single fraction.
For each of the following equations:
Differentiate the function.
State the values of x for which the derivative is zero.
Consider the function y = \dfrac{2 x - 5}{5 x - 2}.
Using the substitution u = 2 x - 5, find u'.
Using the substitution v = 5 x - 2, find v'.
Hence find y'.
Is it possible for the derivative of this function to be zero?
Differentiate the following functions using the quotient rule:
Differentiate y = \dfrac{x^{2}}{x + 3} and find the value of a if y' = 0 at x = a.
Find the value of f' \left( 0 \right) if f \left( x \right) = \dfrac{x}{\sqrt{16 - x^{2}}}.
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.
Find the equation of the normal to the curve y = \left(x^{2} + 1\right)^{4} at the point where x = 1.
Find the values of x such that the gradient of the tangent to the curve y = \dfrac{6 x - 1}{3 x - 1} is - 3.
Consider the function f \left( x \right) = \dfrac{4 x^{9}}{\left(x + 2\right)^{4}}.
Find f' \left( 2 \right).
Is the function increasing or decreasing at x = 2?
The curve y = \sqrt{x - 3} has a tangent with a gradient of \dfrac{1}{2} at the point P. Find the coordinates of P.
f \left( x \right) and g \left( x \right) are differentiable functions such that f \left( x \right) \neq g \left( x \right).
Suppose f \left( 2 \right) = 3, g \left( 2 \right) = 2, f' \left( 2 \right) = 4 and g' \left( 2 \right) = 5.
Find the value of \dfrac{d}{dx} \left( f \left( x \right) g \left( x \right)^{4} \right) at x = 2.
Find the value of \dfrac{d}{dx} \left(\dfrac{g \left( x \right)}{f \left( x \right) + g \left( x \right)}\right) at x = 2.