SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.1 - Differentiation [Latest edition]

If y = sec (tan⁻¹x), then `dy/dx` at x = 1 is ______.

If f(x) = log_x (log x) then f'(e) is ______

If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.

If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______

If y = `tan^-1((2x)/(1 - x^2))`, x ∈ (−1, 1) then `("d"y)/("d"x)` = ______.

If g is the inverse of f and f'(x) = `1/(1 + x^4)` then g'(x) = ______

If sin⁻¹(x³ + y³) = a then `("d"y)/("d"x)` = ______

If x = cos⁻¹(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

If x² + y² = 1, then `(d^2x)/(dy^2)` = ______.

If x² + y² = t + `1/"t"` and x⁴ + y⁴ = t² + `1/"t"^2` then `("d"y)/("d"x)` = ______

If x = a t⁴ y = 2a t² then `("d"y)/("d"x)` = ______

Differentiate y = `sqrt(x^2 + 5)` w.r. to x

Differentiate y = e^tanx w.r. to x

If y = sin⁻¹ (2^x), find `("d"y)/(""d"x)` 

If f(x) is odd and differentiable, then f′(x) is

If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)` 

If y = log [cos(x⁵)] then find `("d"y)/("d"x)`

If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.

Find the derivative of the inverse of function y = 2x³ – 6x and calculate its value at x = −2

Let f(x) = x⁵ + 2x – 3 find (f⁻¹)'(-3)

If y = cos⁻¹ [sin (4^x)], find `("d"y)/("d"x)`

If y = `tan^-1[sqrt((1 + cos x)/(1 - cos x))]`, find `("d"y)/("d"x)`

If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.

Differentiate sin² (sin⁻¹(x²)) w.r. to x

If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`

If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`

Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x

Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x

Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x

If log₅ `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`

If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`

Find the derivative of cos⁻¹x w.r. to `sqrt(1 - x^2)`

If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`

If y = 5^x. x⁵. x^x. 5⁵ , find `("d"y)/("d"x)`

If y = `"e"^("m"tan^(-1)x`, show that `(1 + x^2) ("d"^2y)/("d"x^2) + (2x - "m")("d"y)/("d"x)` = 0

If x⁷ . y⁵ = (x + y)¹², show that `("d"y)/("d"x) = y/x`

Differentiate `tan^-1[(sqrt(1 + x^2) - 1)/x]` w.r. to `tan^-1[(2x sqrt(1 - x^2))/(1 - 2x^2)]`

If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then find `("d"y)/("d"x)`

If y = cos(m cos^–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin²x

Suppose y = f(x) is a differentiable function of x on an interval I and y is one – one, onto and `("d"y)/("d"x)` ≠ 0 on I. Also if f^–1(y) is differentiable on f(I), then `("d"x)/("d"y) = 1/(("d"y)/("d"x)), ("d"y)/("d"x)` ≠ 0

If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost