Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

Differentiate the function with respect to x.

(log x)^x + x^{log x}

Differentiate the function with respect to x.

`(sin x)^x + sin^(-1) sqrtx`

Differentiate the function with respect to x.

x^{sin x} + (sin x)^{cos x}

Differentiate the function with respect to x.

`x^(xcosx) + (x^2 + 1)/(x^2 -1)`

Differentiate the function with respect to x.

`(x cos x)^x + (x sin x)^(1/x)`

Find `bb(dy/dx)` for the given function:

x^y + y^x = 1

Find `bb(dy/dx)` for the given function:

y^x = x^y

Find `bb(dy/dx)` for the given function:

(cos x)^y = (cos y)^x

Find `bb(dy/dx)` for the given function:

xy = `e^((x - y))`

Find the derivative of the function given by f(x) = (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) and hence find f′(1).

Differentiate (x² – 5x + 8) (x³ + 7x + 9) in three ways mentioned below:

1. By using the product rule.
2. By expanding the product to obtain a single polynomial.
3. By logarithmic differentiation.

Do they all give the same answer?

If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.

Differentiate the function with respect to x:

x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.

If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.

For given vectors,  `veca = 2hati - hatj + 2hatk` and `vecb = -hati  + hatj - hatk`, find the unit vector in the direction of the vector `veca +vecb`.

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

`x/a + y/b = 1`