Differentiate each of the following functions. (Do not use the product rule and quotiate rule

Question

a) 2x³ + 3/x³ + 4∛x + 5/∛x

b) f(x) = (3x - 2x√x)(7x + 2)

c) f(x) = (t^2 - 2t + 3) / 2t^2

Answer 1

a) The sum rule for derivatives states that the derivative of a sum is equal to the sum of the derivatives (also true for subtraction). In symbols, this means for a function f(x) = g(x) + h(x), its derivative is f'(x) = g'(x) + h'(x). Basically, take the derivative of each term individually, and add them together.

f(x) = 2x³ + 3/x³ + 4∛x + 5/∛x

Let's rewrite this function such that we can use the power rule to take the derivative. We want to make it easier to visualize what power each of these terms is raised to.

f(x) = 2x³ + 3x^-3 + 4x^1/3 + 5x^-1/3, (use the power rule on each term)

f'(x) = 6x² - 9x^-4 + (4/3)x^-2/3 - (5/3)x^-4/3 (at this point we are technically done, but could simplify further)

f'(x) = 6x² - 9/x⁴ + 4/(3x^2/3) - 5/(3x^4/3)

b) f(x) = (3x - 2x√x)(7x + 2)

Since we are not using the product rule, let's simply multiply this out and distribute the terms.

(Note that x*sqrt(x) is x¹*x^1/2 = x^3/2)

f(x) = 21x² + 6x - 14x^5/2 - 4x^3/2 (now we are in a format similar to part (a), and we can use the power rule from here)

f'(x) = 42x + 6 - 35x^3/2 - 6x^1/2

c) f(t) = (t^2 - 2t + 3) / 2t^2 (problem says "f(x)" but I am calling this f(t), as I assume this is what was intended)

Let's divide each term individually by 2t² and split this into 3 fractions. We can rewrite the function as follows:

f(t) = (t²/2t²) - (2t/2t²) + (3/2t²)

f(t) = 1/2 - (1/t) + (3/2t²)

f(t) = 1/2 - t^-1 + (3/2)t^-2

At this point, we can use the power rule one more time, but I will leave the final result as an exercise after showing how to do the first two parts.