Differentiate both sides with respect to x, treating y as a function of x. d/dx(x^2) + d/dx(y^2) = d/dx(25). This gives 2x + 2y (dy/dx) = 0. Solving for dy/dx: 2y (dy/dx) = -2x, so dy/dx = -2x/(2y) = -x/y.

dy/dx = -x/y

Take the natural logarithm of both sides: ln y = ln(x^x) = x ln x. Now differentiate both sides with respect to x. On the left: (1/y)(dy/dx). On the right, using the product rule on x ln x: (1)(ln x) + x(1/x) = ln x + 1. So (1/y)(dy/dx) = ln x + 1, which gives dy/dx = y(ln x + 1) = x^x (1 + ln x).

dy/dx = x^x (1 + ln x)

First derivative using the product rule: dy/dx = e^x sin x + e^x cos x = e^x(sin x + cos x). Differentiate again using the product rule: d2y/dx2 = e^x(sin x + cos x) + e^x(cos x - sin x) = e^x(sin x + cos x + cos x - sin x) = e^x(2 cos x) = 2 e^x cos x.

d2y/dx2 = 2 e^x cos x

Area A = pi r^2. Differentiate with respect to time t: dA/dt = 2 pi r (dr/dt). Given dr/dt = 3 cm/s and r = 10 cm, so dA/dt = 2 pi (10)(3) = 60 pi cm^2/s. Thus the area is increasing at the rate of 60 pi square cm per second (about 188.5 cm^2/s).

dA/dt = 2 pi r (dr/dt)

Find the derivative: f'(x) = 2x - 4 = 2(x - 2). Set f'(x) = 0 to get the critical point: x = 2. For x < 2, f'(x) < 0, so the function is decreasing on (-infinity, 2). For x > 2, f'(x) > 0, so the function is increasing on (2, infinity). Thus f is decreasing on (-infinity, 2) and increasing on (2, infinity).

The slope of the tangent is dy/dx = 2x. At the point (1, 1), the slope = 2(1) = 2. Using the point-slope form y - y1 = m(x - x1) with m = 2 and (x1, y1) = (1, 1): y - 1 = 2(x - 1), so y - 1 = 2x - 2, giving y = 2x - 1. This is the equation of the tangent.

y - y1 = m(x - x1)

Find the first derivative: f'(x) = 3x^2 - 3 = 3(x^2 - 1). Set f'(x) = 0: x^2 = 1, so x = 1 or x = -1. Second derivative: f''(x) = 6x. At x = 1, f''(1) = 6 > 0, so there is a local minimum; minimum value f(1) = 1 - 3 + 2 = 0. At x = -1, f''(-1) = -6 < 0, so there is a local maximum; maximum value f(-1) = -1 + 3 + 2 = 4.

f'(x) = 0; use f''(x) sign

Let the two numbers be x and 16 - x (since their sum is 16). Their product P = x(16 - x) = 16x - x^2. To maximise, differentiate: dP/dx = 16 - 2x, and set it to zero: 16 - 2x = 0, so x = 8. The second derivative d2P/dx2 = -2 < 0, confirming a maximum. So the numbers are x = 8 and 16 - 8 = 8, and the maximum product is 8 x 8 = 64.

Use the substitution t = 1 + x^2, so dt = 2x dx. The integral becomes the integral of (1/t) dt = ln|t| + C. Substituting back t = 1 + x^2, the answer is ln(1 + x^2) + C. (Note that 1 + x^2 is always positive, so the modulus sign is not needed.)

integral of f'(x)/f(x) dx = ln|f(x)| + C

Integration by parts uses the integral of u v dx = u (integral of v) - integral of (u' times integral of v). Take u = x (so u' = 1) and dv = e^x dx (so integral of dv = e^x). Then integral of x e^x dx = x e^x - integral of (1 x e^x) dx = x e^x - e^x + C = e^x(x - 1) + C.

integral u dv = u v - integral v du

Write 1/((x - 1)(x - 2)) = A/(x - 1) + B/(x - 2). Then 1 = A(x - 2) + B(x - 1). Putting x = 1: 1 = A(-1), so A = -1. Putting x = 2: 1 = B(1), so B = 1. Hence the integral = integral of [-1/(x - 1) + 1/(x - 2)] dx = -ln|x - 1| + ln|x - 2| + C = ln|(x - 2)/(x - 1)| + C.

First find the indefinite integral: the integral of x^2 dx = x^3/3. Now apply the limits using the fundamental theorem of calculus: the integral from 0 to 1 of x^2 dx = [x^3/3] evaluated from 0 to 1 = (1^3/3) - (0^3/3) = 1/3 - 0 = 1/3.

integral(a to b) f(x) dx = F(b) - F(a)

The integral of sin x dx = -cos x. Applying the limits from 0 to pi/2: the integral from 0 to pi/2 of sin x dx = [-cos x] from 0 to pi/2 = (-cos(pi/2)) - (-cos 0) = (-0) - (-1) = 0 + 1 = 1. So the value of the integral is 1.

integral of sin x dx = -cos x

The function f(x) = x^3 is an odd function, because f(-x) = (-x)^3 = -x^3 = -f(x). By the property of definite integrals, the integral from -a to a of an odd function is zero. Therefore the integral from -1 to 1 of x^3 dx = 0. (We can also verify: [x^4/4] from -1 to 1 = 1/4 - 1/4 = 0.)

integral(-a to a) of odd function = 0

The area is given by the definite integral of y with respect to x from 0 to 3: Area = the integral from 0 to 3 of x^2 dx = [x^3/3] from 0 to 3 = (3^3/3) - (0) = 27/3 = 9. So the required area is 9 square units.

Area = integral(a to b) y dx

The area is the definite integral of y = 2x from 0 to 3: Area = the integral from 0 to 3 of 2x dx = [x^2] from 0 to 3 = (3^2) - (0^2) = 9. So the required area is 9 square units. (This agrees with the area of the triangle of base 3 and height 6, which is (1/2)(3)(6) = 9.)

Area = integral(a to b) y dx

The equation x^2 + y^2 = 16 represents a circle centred at the origin with radius r = 4 (since r^2 = 16). The area enclosed by a circle of radius r is pi r^2. So the area = pi (4)^2 = 16 pi square units. (By integration, the area = 4 times the integral from 0 to 4 of sqrt(16 - x^2) dx = 16 pi.)

Area of circle = pi r^2

Separate the variables: dy/(1 + y^2) = dx/(1 + x^2). Now integrate both sides: the integral of dy/(1 + y^2) = the integral of dx/(1 + x^2). This gives tan inverse (y) = tan inverse (x) + C, where C is the constant of integration. This is the general solution of the differential equation.

integral dy/(1 + y^2) = tan inverse (y)

Separate the variables by cross-multiplying: y dy = x dx. Integrate both sides: the integral of y dy = the integral of x dx, giving y^2/2 = x^2/2 + C1. Multiplying by 2: y^2 = x^2 + C (where C = 2 C1). So the general solution is y^2 - x^2 = C, which represents a family of hyperbolas.

A linear differential equation of the first order and first degree has the standard form dy/dx + P y = Q, where P and Q are functions of x only (or constants). It is solved using the integrating factor (IF) = e^(integral of P dx). Multiplying the equation by the integrating factor makes the left side an exact derivative, and the solution is y x (IF) = integral of [Q x (IF)] dx + C.

IF = e^(integral of P dx)