Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 - Differentiation [Latest edition]

Find `"dy"/"dx"` if, y = `sqrt("x" + 1/"x")`

Find `"dy"/"dx"` if, y = `root(3)("a"^2 + "x"^2)`

Find `"dy"/"dx"` if, y = (5x³ - 4x² - 8x)⁹ 

Find `"dy"/"dx"` if, y = log(log x)

Find `"dy"/"dx"` if, y = log(10x⁴ + 5x³ - 3x² + 2)

Find `"dy"/"dx"` if, y = log(ax² + bx + c) 

Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`

Find `"dy"/"dx"` if, y = `"a"^((1 + log "x"))`

Find `"dy"/"dx"` if, y = `5^(("x" + log"x"))`

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x² 

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 18x + log(x - 4).

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25x + log(1 + x²)

Find the marginal demand of a commodity where demand is x and price is y.

y = `"x"*"e"^-"x" + 7`

Find the marginal demand of a commodity where demand is x and price is y.

y = `("x + 2")/("x"^2 + 1)`

Find the marginal demand of a commodity where demand is x and price is y.

y = `(5"x" + 9)/(2"x" - 10)`

Find `"dy"/"dx"`if, y = `"x"^("x"^"2x")`

Find `"dy"/"dx"`if, y = `"x"^("e"^"x")`

Find `"dy"/"dx"`if, y = `"e"^("x"^"x")`

Find `"dy"/"dx"`if, y = `(1 + 1/"x")^"x"`

Find `"dy"/"dx"`if, y = (2x + 5)^x 

Find `"dy"/"dx"`if, y = `root(3)(("3x" - 1)/(("2x + 3")(5 - "x")^2))`

Find `"dy"/"dx"`if, y = `(log "x"^"x") + "x"^(log "x")`

Find `"dy"/"dx"`if, y = `("x")^"x" + ("a"^"x")`

Find `"dy"/"dx"`if, y = `10^("x"^"x") + 10^("x"^10) + 10^(10^"x")`

Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`

Find `"dy"/"dx"` if, x³ + y³ + 4x³y = 0 

Find `"dy"/"dx"` if, x³ + x²y + xy² + y³ = 81

Find `"dy"/"dx"` if, ye^x + xe^y = 1 

Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`

Find `"dy"/"dx"` if, xy = log (xy)

Solve the following:

If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`

Solve the following:

If log (x + y) = log (xy) + a then show that, `"dy"/"dx" = (- "y"^2)/"x"^2`.

Solve the following:

If `"e"^"x" + "e"^"y" = "e"^("x + y")` then show that, `"dy"/"dx" = - "e"^"y - x"`.

Find `"dy"/"dx"`, if x = at², y = 2at

Find `"dy"/"dx"`, if x = 2at² , y = at⁴

Find `"dy"/"dx"`, if x = e^3t , y = `"e"^(4"t" + 5)`

Find `"dy"/"dx"`, if x = `("u" + 1/"u")^2, "y" = (2)^(("u" + 1/"u"))`

Find `"dy"/"dx"`, if x = `sqrt(1 + "u"^2), "y" = log (1 + "u"^2)`

Find `"dy"/"dx"`, if Differentiate 5^x with respect to log x

Solve the following.

If x = `"a"(1 - 1/"t"), "y" = "a"(1 + 1/"t")`, then show that `"dy"/"dx" = - 1`

Solve the following.

If x = `"4t"/(1 + "t"^2), "y" = 3((1 - "t"^2)/(1 + "t"^2))` then show that `"dy"/"dx" = (-9"x")/"4y"`.

Solve the following.

If x = t . log t, y = t^t, then show that `"dy"/"dx" - "y" = 0`

Find `("d"^2"y")/"dx"^2`, if y = `sqrt"x"`

Find `("d"^2"y")/"dx"^2`, if y = `"x"^5`

Find `("d"^2"y")/"dx"^2`, if y = `"x"^-7`

Find `("d"^2"y")/"dx"^2`, if y = `"e"^"x"`

Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`

Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`

Choose the correct alternative.

If y = (5x³ - 4x² - 8x)⁹, then `"dy"/"dx"` =

Choose the correct alternative.

If y = `sqrt("x" + 1/"x")`, then `"dy"/"dx" = ?`

Choose the correct alternative.

If y = `"e"^(log "x")`, then `"dy"/"dx" = ?`

Choose the correct alternative.

If y = 2x² + 2² + a², then `"dy"/"dx" = ?`

Choose the correct alternative.

If y = 5^x . x⁵, then `"dy"/"dx" = ?` 

Choose the correct alternative.

If y = log `("e"^"x"/"x"^2)`, then `"dy"/"dx" = ?` 

Choose the correct alternative.

If ax² + 2hxy + by² = 0 then `"dy"/"dx" = ?` 

Choose the correct alternative.

If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ? 

Choose the correct alternative.

If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2`  then `"dy"/"dx"` = ? 

Choose the correct alternative.

If x = 2at² , y = 4at, then `"dy"/"dx" = ?`

Fill in the Blank

If 3x²y + 3xy² = 0, then `"dy"/"dx"` = ________

Fill in the Blank

If `"x"^"m"*"y"^"n" = ("x + y")^("m + n")`, then `"dy"/"dx" = square/"x"`

Fill in the Blank

If 0 = log(xy) + a, then `"dy"/"dx" =  (-"y")/square`

Fill in the blank.

If x = t log t and y = t^t, then `"dy"/"dx"` = ____

Fill in the blank.

If y = x . log x, then `("d"^2"y")/"dx"^2`= _____

Fill in the blank.

If y = `[log ("x")]^2  "then" ("d"^2"y")/"dx"^2 =` _____

Fill in the blank.

If x = `"y" + 1/"y"`, then `"dy"/"dx" =`____

Fill in the blank.

If y = `"e"^"ax"`, then `"x" * "dy"/"dx" =`____

Fill in the blank.

If x = t log t and y = t^t, then `"dy"/"dx"` = ____

Fill in the blank.

If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______

State whether the following is True or False:

If f′ is the derivative of f, then the derivative of the inverse of f is the inverse of f′.

State whether the following is True or False:

The derivative of `log_"a""x"`, where a is constant is `1/("x"*log"a")`.

State whether the following is True or False:

The derivative of f(x) = a^x, where a is constant is x.a^x-1.

State whether the following is True or False:

The derivative of polynomial is polynomial.

State whether the following is True or False:

`"d"/"dx"(10^"x") = "x"*10^("x" - 1)`

State whether the following is True or False:

If y = log x, then `"dy"/"dx" = 1/"x"`

State whether the following is True or False:

If y = e², then `"dy"/"dx" = 2"e"`

State whether the following is True or False:

The derivative of a^x is a^x . loga.

State whether the following is True or False:

The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`

Solve the following:

If y = (6x³ - 3x² - 9x)¹⁰, find `"dy"/"dx"` 

Solve the following:

If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"` 

Solve the following:

If y = [log(log(logx))]², find `"dy"/"dx"`

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x  – x².

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5"x" + 7)/("2x" - 13)`.

Find `"dy"/"dx"`, if y = x^x.

Find `"dy"/"dx"`, if y = `2^("x"^"x")`.

Find `"dy"/"dx"` if y = `sqrt(((3"x" - 4)^3)/(("x + 1")^4("x + 2")))`

Find `"dy"/"dx"` if y = `"x"^"x" + ("7x" - 1)^"x"`

If y = `"x"^3 + 3"xy"^2 + 3"x"^2"y"` Find `"dy"/"dx"`

If `"x"^3 + "y"^2 + "xy" = 7` Find `"dy"/"dx"`

If `"x"^3"y"^3 = "x"^2 - "y"^2`, Find `"dy"/"dx"`

If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`

If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`

Find `"dy"/"dx"` if x = 5t², y = 10t.  

Find `"dy"/"dx"` if x = `"e"^"3t",  "y" = "e"^(sqrt"t")`.

Differentiate log (1 + x²) with respect to a^x.

Differentiate `"e"^("4x" + 5)` with respect to 10^4x.

Find `("d"^2"y")/"dx"^2`, if y = log (x).

Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at²

Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`

If x² + 6xy + y² = 10, then show that `("d"^2"y")/"dx"^2 = 80/("3x" + "y")^3`.

If ax² + 2hxy + by² = 0, then show that `("d"^2"y")/"dx"^2` = 0