Questions & Answers: "Differenciation"

To find the derivative of f(x) = 1/√x, we first rewrite it in power form as f(x) = x^(-1/2). Applying the power rule (d/dx [x^n] = n*x^(n-1)), we get f'(x) = (-1/2) * x^(-1/2 - 1) = (-1/2) * x^(-3/2) = -1 / (2 * x^(3/2)). Now, substitute x = 4: f'(4) = -1 / (2 * 4^(3/2)). Since 4^(3/2) = (√4)^3 = 2^3 = 8, the derivative is -1 / (2 * 8) = -1/16. Option B is incorrect due to the sign; options C and D represent incorrect power rule applications.

We use the Quotient Rule: d/dx [u/v] = (vdu/dx - udv/dx) / v². Here, u = x and v = sin(x). Therefore, du/dx = 1 and dv/dx = cos(x). Substituting these into the formula: dy/dx = (sin(x) * 1 - x * cos(x)) / (sin(x))² = (sin(x) - x*cos(x)) / sin²(x). Option B reverses the order of the numerator terms (leading to a sign error), Option C incorrectly uses addition instead of subtraction, and Option D is a common mistake where students differentiate numerator and denominator independently (violating the quotient rule).

According to the chain rule, d/dx [f(g(x))] = f'(g(x)) * g'(x). Here, let f(u) = cos(u) where u = x² + 5. The derivative of the outer function f(u) is -sin(u), and the derivative of the inner function u = x² + 5 is 2x. Multiplying them together, we get dy/dx = -sin(x² + 5) * (2x) = -2x * sin(x² + 5). Option A misses the negative sign required for the derivative of cosine. Option C forgets to differentiate the inner function (the 2x term). Option D incorrectly keeps the cosine function instead of changing it to sine.

By applying the Product Rule [d/dx(uv) = u'v + uv'], where u = x and v = sin(x). Here u' = 1 and v' = cos(x). Thus, f'(x) = (1)sin(x) + (x)cos(x). Option A and D are incorrect because they only differentiated one part of the product. Option C is incorrect because it uses the wrong sign (subtraction instead of addition).

The derivative is the instantaneous rate of change, defined by the limit of the difference quotient as the increment 'h' approaches zero. Option A is the standard mathematical definition. Option B is incorrect because it uses f(x-h) - f(x) which would lead to the negative of the derivative unless the denominator was also -h. Option C is incorrect because the numerator must represent the change in y (difference), not the sum. Option D is incorrect because h must approach 0, not infinity, to find the slope at a specific point.

Using the Chain Rule, we differentiate the outer function sin(u) to get cos(u), and then multiply by the derivative of the inner function (ax + b), which is 'a'. Thus, result = a * cos(ax + b). Option A misses the inner derivative 'a'. Option C incorrectly adds a negative sign (derivative of sine is positive cosine). Option D fails to differentiate the sine function.

Using the power rule d/dx(x^n) = nx^(n-1) and the rule that the derivative of a constant is 0: d/dx(x^3) = 3x^2, d/dx(-2x) = -2, and d/dx(10) = 0. Adding these gives 3x^2 - 2. Option A incorrectly kept the constant 10. Option C used the wrong coefficient for x^2. Option D incorrectly differentiated -2x.

The Quotient Rule states d/dx(u/v) = (u'v - uv') / v^2. Here u = x+1, v = x-1, u' = 1, v' = 1. So, f'(x) = [1*(x-1) - (x+1)*1] / (x-1)^2 = (x - 1 - x - 1) / (x-1)^2 = -2 / (x-1)^2. Option B has the wrong sign. Option C is missing the factor of 2. Option D uses the wrong denominator (x+1 instead of x-1).

The slope of the tangent is the value of the derivative at that point. The derivative of y = x^2 is dy/dx = 2x. Evaluating at x = 3, we get 2(3) = 6. Option A is the x-value. Option B is the y-value (x^2). Option D is the derivative function's coefficient.

The derivative of sec(x) is a standard trigonometric derivative derived from 1/cos(x) using the quotient rule, resulting in sec(x)tan(x). Option A is incorrect as it's unrelated. Option C is incorrect because the derivative of sec(x) is positive (unlike csc(x)). Option D is the derivative of tan(x), not sec(x).

f(x) = x^(-2). Applying the power rule: d/dx(x^-2) = -2 * x^(-2-1) = -2 * x^(-3) = -2 / x^3. Option B is missing the negative sign required by the power rule. Option C and D represent incorrect power subtractions.

Even though it looks like a trigonometric function, sin(pi/3) is a constant value (sqrt(3)/2). The derivative of any constant with respect to x is always 0. Option A is a common trap where students differentiate the sine part without realizing it's a constant. Options B and D are just values of trig functions, not derivatives.

f(x) can be written as x^(1/2). Applying the power rule: d/dx(x^n) = n*x^(n-1), we get (1/2)*x^(1/2 - 1) = (1/2)x^(-1/2) = 1 / (2sqrt(x)). Option B is the reciprocal. Option C is missing the factor of 1/2. Option D incorrectly includes a negative sign.