Question

Differentiate the following function using a calculator.

a) f(x) = x^3 - 2x^2 + 4x - 1
b) g(x) = e^(2x) + ln(x)
c) h(x) = sin(3x) + cos(2x)
d) k(x) = √x + 1/x

Answer 1

To differentiate the given functions, we use the power rule, product rule, chain rule, and logarithmic differentiation. Step by step, the derivatives are a) 3x^2 - 4x + 4, b) 2e^(2x) + 1/x, c) 3cos(3x) - 2sin(2x), and d) 1/(2√x) - 1/x^2.

Step-by-step explanation:

To differentiate the given functions, we can use the power rule, product rule, chain rule, and logarithmic differentiation. Let's differentiate each function step by step:

a) f(x) = x^3 - 2x^2 + 4x - 1:

Using the power rule, we differentiate each term separately. The derivative of x^3 is 3x^2, the derivative of -2x^2 is -4x, the derivative of 4x is 4, and the derivative of -1 is 0. Therefore, the derivative of f(x) is 3x^2 - 4x + 4.

b) g(x) = e^(2x) + ln(x):

Using the chain rule, the derivative of e^(2x) is 2e^(2x), and the derivative of ln(x) is 1/x. Therefore, the derivative of g(x) is 2e^(2x) + 1/x.

c) h(x) = sin(3x) + cos(2x):

Using the derivatives of trigonometric functions, the derivative of sin(3x) is 3cos(3x), and the derivative of cos(2x) is -2sin(2x). Therefore, the derivative of h(x) is 3cos(3x) - 2sin(2x).

d) k(x) = √x + 1/x:

Using the power rule and the chain rule, the derivative of √x is 1/(2√x), and the derivative of 1/x is -1/x^2. Therefore, the derivative of k(x) is 1/(2√x) - 1/x^2.