Praxis Math Derivatives and Integration Quiz

To find the derivative of the function f(x) = 3x⁴ − 2x² + 5, apply the power rule. The derivative of 3x⁴ is 12x³, and the derivative of -2x² is -4x. The constant term 5 has a derivative of 0. Thus, combining these results gives the derivative as 12x³ − 4x.

To find the derivative of g(x) = (2x + 1)⁵ using the chain rule, first differentiate the outer function, which gives 5(2x + 1)⁴. Then, multiply by the derivative of the inner function, 2. This results in 5(2)(2x + 1)⁴ = 10(2x + 1)⁴.

To find the derivative of h(x) = eˣ sin(x), apply the product rule, which states that the derivative of two functions multiplied together is the derivative of the first times the second plus the first times the derivative of the second. This results in h'(x) = eˣ sin(x) + eˣ cos(x), which can be factored to eˣ(sin(x) + cos(x)).

To evaluate the integral ∫ 5x³ dx, we apply the power rule of integration, which states that ∫ x^n dx = (x^(n+1))/(n+1) + C. Here, n is 3, so we increase the exponent to 4 and divide by 4, resulting in (5x⁴)/4 + C.

To solve the integral ∫ (3x² + 2x − 1) dx, apply the power rule of integration. Each term is integrated separately: the integral of 3x² is x³, the integral of 2x is x², and the integral of -1 is -x. Thus, combining these results gives x³ + x² - x + C.

Using u-substitution, let \( u = x^2 + 1 \). Then, \( du = 2x \, dx \), allowing us to rewrite the integral as \( \int u^3 \, du \). Evaluating this gives \( \frac{u^4}{4} + C \). Substituting back \( u = x^2 + 1 \) results in \( \frac{(x^2 + 1)^4}{4} + C \), which simplifies to \( \frac{(x^2 + 1)^4}{2} + C \).

A constant function does not change regardless of the input value, meaning its rate of change is always zero. The derivative measures this rate of change; therefore, for any constant function, the derivative will consistently yield zero, confirming that the statement is true.

To differentiate the function f(x) = x² ln(x), the product rule is appropriate because it involves the multiplication of two functions: x² and ln(x). The product rule states that the derivative of a product of two functions is the derivative of the first function times the second, plus the first function times the derivative of the second.

To find the critical points of the function \( f(x) = x^3 - 3x^2 + 2 \), we first compute the derivative \( f'(x) \). Setting \( f'(x) = 0 \) allows us to solve for \( x \). The solutions \( x = 0 \) and \( x = 2 \) indicate where the function's slope is zero, identifying critical points.

To evaluate the definite integral ∫₀¹ 4x³ dx, we first find the antiderivative of 4x³, which is x⁴. Then, we apply the Fundamental Theorem of Calculus by calculating the difference between the values at the upper and lower limits: (1)⁴ - (0)⁴ = 1 - 0 = 1. Thus, the result of the integral is 1.

The integral of eˣ with respect to x is indeed eˣ + C, where C is the constant of integration. This is because the derivative of eˣ is itself, making it one of the unique functions in calculus that retains its form under differentiation and integration. Thus, the statement is true.

To find the derivative of f(x) = ln(x) / x using the quotient rule, we differentiate the numerator and denominator separately. The quotient rule states that if f(x) = u/v, then f'(x) = (u'v - uv') / v². Applying this, we obtain (1 - ln(x)) / x² as the derivative.