Exercises
Sharpen your differential calculus knowledge with this quiz on derivatives and differentiation rules. Solve problems involving polynomial functions, first and second derivatives, exponential and logarithmic functions, and trigonometric derivatives. Practice applying the product rule and chain rule, evaluating derivatives at specific points, and using the limit definition of the derivative. Questions cover expressions such as 3x², x³ - 4x + 6, ln(x), eˣ, sin(x), cos(2x), and composite functions like (3x - 4)⁵. Ideal for students reviewing introductory calculus concepts and preparing for exams.
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To find the derivative of f(x) = 3x^2, apply the power rule: differentiate x^n to get n*x^(n-1). Here, f'(x) = 2*3*x^(2-1) = 6x. At x = 2, f'(2) = 6*2 = 12.
The derivative of a function g(x) is found using the power rule, which states that the derivative of x^n is n*x^(n-1). Applying this to g(x) = x^3 - 4x + 6, we differentiate each term:
So, g'(x) = 3x^2 - 4. Therefore, the correct option is 1.
To find the second derivative, we first find the first derivative of h(x) = 2x^4 - 3x^2 + x. The first derivative h'(x) is:
h'(x) = 8x^3 - 6x + 1.
Now, we find the second derivative h''(x) by differentiating again:
h''(x) = 24x^2 - 6.
Therefore, the second derivative of h(x) is 24x^2 - 6, corresponding to Option 1.
To find the derivative of the function f(x) = 1/x at x = 1, first compute the derivative, which is f'(x) = -1/x². Substituting x = 1 into f'(x) gives f'(1) = -1/1² = -1. Therefore, the true statement is that f'(x) = -1 at x = 1.
The derivative of the exponential function f(x) = e^x is e^x. This is a unique property of the exponential function with base e, also known as Euler's number. It means the rate of change of the function is equal to the value of the function itself at any point.
The Product rule is typically applied to differentiate the product of two functions. It states that if you have two differentiable functions u(x) and v(x), their product u(x)v(x) is differentiated as u'(x)v(x) + u(x)v'(x).
Using the definition of the derivative involves evaluating the limit: lim (h → 0) [(ln(x + h) - ln(x))/h]. By applying the properties of logarithms, this becomes lim (h → 0) [ln((x + h)/x)/h] = lim (h → 0) [ln(1 + h/x)/h]. As h approaches 0, this approaches 1/x, which is the derivative of ln(x). Thus, the correct answer is 1/x.
The function f(x) = sin(x) has a first derivative of f'(x) = cos(x) and a second derivative of f''(x) = -sin(x). Thus, the correct option is 3) -sin(x).
The function y = cos(2x) is a composition of functions. To find the derivative dy/dx, we apply the chain rule. The derivative of cos(u) is -sin(u), and for u = 2x, we also need to multiply by the derivative of 2x which is 2. Therefore, dy/dx = -sin(2x) * 2 = -2sin(2x).
The function f(x) = (3x - 4)^5 requires the chain rule for differentiation. The outer function is u^5 where u = (3x - 4). The derivative of u^5 is 5u^4. The derivative of the inner function (3x - 4) is 3. Applying the chain rule: f'(x) = 5(3x - 4)^4 * 3 = 15(3x - 4)^4. Thus, the correct option is 1.
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