11.09 Product Rule
To differentiate y = x^{6} \left(x^{4} + 4\right) using the product rule, let u = x^{6} and v = x^{4} + 4, then:
Find u'.
Find v'.
Hence, find \dfrac{dy}{dx}.
To differentiate y = \left(x^{7} + 5\right) \left(x^{6} + 6\right) using the product rule, let u = x^{7} + 5 and v = x^{6} + 6, then:
Find \dfrac{d u}{d x}.
Find \dfrac{d v}{d x}.
Hence, find \dfrac{dy}{dx}.
Consider the function y = x^{3} \left(x^{2} + 9\right).
Differentiate y by first expanding the brackets.
Differentiate y using the product rule, letting u = x^{3} and v = x^{2} + 9.
Differentiate the following functions:
f \left( x \right) = \left( 3 x - 2\right) \left( 4 x - 5\right)
g \left( t \right) = \left( 2 t^{3} - 3\right) \left(3 - t\right)
g \left( y \right) = \left( 7 y^{4} - y^{2}\right) \left(y^{2} - 5\right)
f \left( x \right) = \left(x^{\frac{4}{3}} + 6 \sqrt{x}\right) \left( 6 x + 3\right)
f \left( x \right) = x \sqrt{3 - x}
f \left( x \right) = \sqrt[3]{x^{2}} \left( 2 x - x^{2}\right)
f \left( x \right) = x^{\frac{1}{3}} \left(1 - x\right)^{\frac{2}{3}}
y = \left(2 + \sqrt{x}\right) \left(6 - x^{2}\right)
y = \left(1 + \dfrac{1}{x}\right) \left(3 + x - x^{2}\right)
y = x^{3} \left( 5 x + 3\right)^{7}
y = 6 x^{5} \left(x^{2} + 3\right)^{3}
y = 3 x \left(x^{2} + x + 1\right)^{9}
y = \left( 8 x - 9\right)^{5} \left( 5 x + 7\right)^{7}
y = \left( 3 x + 2\right) \sqrt{5 + 4 x}
y = 8 x \left(5 + 8 x\right)^{\frac{7}{4}} - 3
y = 8 x^{5} \sqrt{ 8 x + 3}
y = 6 x \sqrt{x + 1}
y = - 4 x \sqrt{1 - 2 x}
Consider the function y = \left( 4 x - 3\right) \left( 5 x - 2\right).
Differentiate y.
Hence, differentiate f \left( x \right) = x^{3} \left( 4 x - 3\right) \left( 5 x - 2\right).
For each of the following functions:
Identify possible factors u and v for the function.
Differentiate the function. Give your answer in factorised form.
State the values of x for which the derivative is zero.
The derivative of f \left( x \right) = \left( 3 x^{n} + 4\right) \left( 5 x^{2} - 2 x\right) is of degree 5. Find the value of n.
Consider the function f \left( x \right) = \left(x^{2} - 3 x\right) \left( 2 x - 5\right).
Find f \left( 3 \right).
Find f' \left( 0 \right).
Find f' \left( - 3 \right).
Consider the function g \left( x \right) = x^{3} f \left( x \right), where f \left( x \right) is a function of x. Given that f \left( 3 \right) = 1 and f' \left( 3 \right) = - 3, find g' \left( 3 \right).
Consider the function f \left( x \right) = \left(x + 1\right) \left(x + 3\right)^{3}.
Find f' \left( x \right) in factorised form.
Find the equation of the tangent at \left( - 1 , 0\right).
Find the equation of the normal at \left( - 1 , 0\right).
Find the values of x such that the gradient of the tangent to the curve y = 2 x \left(x + 3\right)^{2} is equal to 14.
Find the gradient of the tangent to the curve y = x \sqrt{ 2 x + 5} at the point where x = 2.