Question

Determine the derivatives using shortcuts of each of the following functions. Do NOT simplify 3 marks each

a. f(x)=3 x⁴ - {2}{x²}+ √{3 x}-7
b. g(x)=(x-2) [(3 x+1)² (5 x-1)

Answer 1

To determine the derivatives of the given functions, we can use the shortcut rules of differentiation. For f(x) = 3x⁴ - 2x² + √(3x) - 7, the derivative is f'(x) = 12x³ - 4x + (3/2√(3x)). For g(x) = (x-2)[(3x+1)²(5x-1)], the derivative is g'(x) = 1 * [(3x+1)²(5x-1)] + (x-2) * [2(3x+1)*(3)*(5) + (3x+1)²(5)].

Step-by-step explanation:

To determine the derivatives of the given functions, we can use the shortcut rules of differentiation. Let's start with function (a): f(x) = 3x⁴ - 2x² + √(3x) - 7.

1. To find the derivative of the first term, 3x⁴, we bring down the exponent and multiply it by the coefficient: 4 * 3 = 12. So the derivative of 3x⁴ is 12x³.
2. For the second term, -2x², we again bring down the exponent and multiply it by the coefficient: 2 * -2 = -4. So the derivative of -2x² is -4x.
3. The derivative of √(3x) can be found using the power rule for fractional exponents. Since the square root can be written as (3x)^(1/2), we bring down the exponent and multiply it by the coefficient: (1/2) * 3 = 3/2. So the derivative of √(3x) is (3/2)(3x)^(1/2 - 1) = (3/2)(3x)^(-1/2) = (3/2√(3x)).
4. Lastly, the derivative of the constant term, -7, is zero.

Putting it all together, the derivative of f(x) = 3x⁴ - 2x² + √(3x) - 7 is f'(x) = 12x³ - 4x + (3/2√(3x)).

For function (b), g(x) = (x-2)[(3x+1)²(5x-1)], we can use the product rule for differentiation. We differentiate the first term (x-2), keep the second term [(3x+1)²(5x-1)] as is, and add the reverse: differentiate the second term and keep the first term as is.

Let's break it down step-by-step:

1. The derivative of the first term, (x-2), is 1.
2. For the second term [(3x+1)²(5x-1)], we use the chain rule: differentiate the outer function, which is the square of a binomial, and multiply it by the derivative of the inner function. The derivative of (3x+1)² is 2(3x+1)*(3), and the derivative of (5x-1) is 5.
3. Then, we add the reverse: differentiate the second term, which is (3x+1)²(5x-1), and keep the first term, (x-2), as is.

Putting it all together, the derivative of g(x) = (x-2)[(3x+1)²(5x-1)] is g'(x) = 1 * [(3x+1)²(5x-1)] + (x-2) * [2(3x+1)*(3)*(5) + (3x+1)²(5)].