Final answer:
To find the derivatives of the given functions: a. Apply the power rule and constant rule to differentiate each term, then combine the derivatives. b. Use the product rule to differentiate the function. c. Differentiate each term and combine the derivatives. d. Apply the product rule to differentiate the function. e. Use the quotient rule to differentiate the function.
Step-by-step explanation:
a. To find the derivative of the function f(x) = 4x⁴ + 2x³ − 8x + 15, you need to differentiate each term. The derivative of 4x⁴ is 16x³, the derivative of 2x³ is 6x², the derivative of -8x is -8, and the derivative of 15 is 0. Combine these derivatives to get the derivative of the function.
b. To find the derivative of the function f(x) = 5x²cos(x), you need to use the product rule. Take the derivative of 5x², which is 10x, and multiply it by cos(x). Then, take the derivative of cos(x), which is -sin(x), and multiply it by 5x². Combine these two terms to get the derivative of the function.
c. To find the derivative of the function f(x) = 5sin(x) - e⁵ˣ, you need to differentiate each term. The derivative of 5sin(x) is 5cos(x), and the derivative of e⁵ˣ is 5e⁵ˣ. Combine these derivatives to get the derivative of the function.
d. To find the derivative of the function f(x) = e³ˣsin(x²), you need to use the product rule. Take the derivative of e³ˣ, which is 3e³ˣ, and multiply it by sin(x²). Then, take the derivative of sin(x²), which is 2xcos(x²), and multiply it by e³ˣ. Combine these two terms to get the derivative of the function.
e. To find the derivative of the function f(x) = (x² - 2)/(x³ + 3x - 2), you need to use the quotient rule. Take the derivative of the numerator, which is 2x, and multiply it by the denominator. Then, take the derivative of the denominator, which is 3x² + 3, and multiply it by the numerator. Subtract these two terms and divide by the square of the denominator to get the derivative of the function.