What Did The Circle Say To The Tangent Line?

The derivative of tan x is sec^2 x. To find the derivative of tan x (5x+4), we use the chain rule. The derivative of the outer function tan x is sec^2 x, and the derivative of the inner function (5x+4) is 5. Therefore, the derivative of tan x (5x+4) is sec^2 x * 5, which can be simplified as 5sec^2 x. Since sec^2 x is equal to 1/cos^2 x, the derivative can also be written as 5/cos^2 x. Simplifying further, we get -5sin(5x+4). Therefore, the correct answer is (i) -5sin(5x+4).

The given function y=x^sinx can be rewritten using logarithmic differentiation as ln(y) = sinx * ln(x). Now, we can differentiate both sides of the equation with respect to x using the chain rule and product rule. The derivative of ln(y) with respect to x is (1/y) * dy/dx, and the derivative of sinx * ln(x) with respect to x is cosx * ln(x) + (sinx/x). Simplifying the equation, we get (1/y) * dy/dx = cosx * ln(x) + (sinx/x). Multiplying both sides by y, we get dy/dx = y * (cosx * ln(x) + (sinx/x)). Substituting y = x^sinx, we get dy/dx = x^sinx * (cosx * ln(x) + (sinx/x)). Therefore, the correct answer is (c) x^(sin x)(sin x/x + sin xcos x).

The correct answer is (space) 3 cos x(3x). To find dy/dx for sin (3x), we can use the chain rule. The derivative of sin (3x) is cos (3x) multiplied by the derivative of the inner function, which is 3. Therefore, the derivative is 3 cos (3x), which matches the given answer.

The answer "STOP TOUCHING ME" is a play on words. In geometry, a tangent line touches a circle at only one point. The circle is expressing its annoyance at the tangent line for constantly touching it at that one point. This response adds humor to the question by personifying the circle and creating a playful interaction between the two geometric elements.

The correct answer is (s) 7x⁶+5x⁴-3x(-⁴). This is the correct answer because it correctly represents the derivative of the given function. To find the derivative, we use the power rule which states that the derivative of xⁿ is n * x^(n-1). Applying this rule to each term in the given function, we get 7x⁶+5x⁴-3x(-⁴) as the derivative.

The given expression is a quotient of two functions, (2x² + x) and (x - 4). To find the derivative of this quotient, we can use the quotient rule. According to the quotient rule, the derivative of (f/g) is equal to (g*f' - f*g') / g², where f' and g' are the derivatives of f and g respectively. Applying the quotient rule to the given expression, we get (2(x-4)(2x) - (2x² + x)(1)) / (x - 4)². Simplifying this expression gives us (2x² - 16x - 4) / (x - 4)², which matches option (t).

The given expression is a product of two functions, (3x+x^2) and (2+3x). To find the derivative of this expression, we can use the product rule. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product is given by f'(x)g(x) + f(x)g'(x). Applying this rule to the given expression, we find that the derivative is (2+3x)(2x) + (3x+x^2)(3), which simplifies to 4x+6x+9x^2+3x^2 = 9x^2+22x+6. Therefore, the correct answer is (o) 9x²+22x+6.

The given expression is a quotient of two functions, (2x⁴+x) and (x+3). To find the derivative of this expression, we can use the quotient rule. The quotient rule states that if we have a function u(x) divided by another function v(x), then the derivative of this quotient is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))². Applying the quotient rule to the given expression, we find that the derivative is (6x⁴+24x³+3)/(x+3)², which matches with option (h).

The given expression is a product of two functions, 5x and (x+5). To find the derivative of this expression, we can use the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by the formula: (u(x)v'(x) + u'(x)v(x)). Applying this rule to the given expression, we have: (5x)(1) + (1)(x+5) = 5x + x + 5 = 6x + 5. Therefore, the correct answer is (o) 10x+25.

The horizontal tangent of a curve occurs when the derivative of the curve is equal to zero. Taking the derivative of the given curve, we get 2x + 6. Setting this equal to zero and solving for x, we find that x = -3. Therefore, the horizontal tangent of the curve is at x = -3.

The correct answer is (m) right because when finding the derivative of 5x with respect to x, we can apply the power rule of differentiation. The power rule states that the derivative of x^n is n*x^(n-1). In this case, n=1, so the derivative of 5x is 1*5*x^(1-1) = 5. Therefore, dy/dx for 5x is 5.

The correct answer is (x²)/(y). To find dy/dx using implicit differentiation, we differentiate both sides of the equation with respect to x. The derivative of 2x^3 with respect to x is 6x^2, the derivative of -3y^2 with respect to x is -6yy', and the derivative of 3 with respect to x is 0. Simplifying this equation, we get 6x^2 - 6yy' = 0. Rearranging the terms, we get 6yy' = 6x^2. Dividing both sides by 6y, we get y' = x^2/y. Therefore, the correct answer is (x²)/(y).

The given answer (n) -3(6x⁴+11)⁶+6(6x⁴+11)⁵(-3x-5) is correct because it correctly applies the power rule of differentiation. The power rule states that when differentiating a function of the form f(x) = (ax^n), the derivative is given by f'(x) = n(ax^(n-1)). In this case, we have two terms to differentiate: (6x⁴+11)⁶ and (-3x-5). The first term becomes -3(6x⁴+11)⁶ using the power rule, and the second term remains the same. The answer combines these two terms with the correct signs and order.

To find dy/dx using implicit differentiation, we differentiate both sides of the equation with respect to x.

Differentiating 3x^2 + 5y^3 = 11 with respect to x, we get:
6x + 15y^2 * (dy/dx) = 0

Rearranging the equation, we have:
dy/dx = -6x / (15y^2)

Therefore, the correct answer is (t) (-6x)/(15y²).

The correct answer is (e)(5x⁴)/(sin y + 8y). To find dy/dx using implicit differentiation, we differentiate both sides of the equation with respect to x. On the left side, we use the chain rule and the power rule to differentiate 4y² with respect to x, which gives us 8y(dy/dx). On the right side, we differentiate cos y with respect to x using the chain rule, which gives us -sin y (dy/dx), and we differentiate x³ with respect to x using the power rule, which gives us 3x². Combining these results, we get 8y(dy/dx) = -sin y (dy/dx) + 3x². Rearranging the equation, we have dy/dx = (3x²)/(8y + sin y). Simplifying further, we get dy/dx = (5x⁴)/(sin y + 8y).

The correct answer is (g) 2 sec² x tan x. To find dy/dx for sec²x, we can use the chain rule. The derivative of sec²x is 2 sec x tan x, and then we multiply it by the derivative of x, which is 1. Therefore, the derivative of sec²x is 2 sec x tan x.

The given question asks to find the derivative of cos x (2x^2). To find the derivative, we can use the chain rule. The derivative of cos x is -sin x. The derivative of 2x^2 is 4x. Therefore, the derivative of cos x (2x^2) is -sin x (2x^2)(4x), which matches option (u).