Advertisements
Advertisements
प्रश्न
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
Advertisements
उत्तर
We may rewrite f as f(x) = `{{:(x^2",", "if" x ≥ 0),(-x^2",", "if" x < 0):}`
Now Lf ′(0) = `lim_("h" -> 0^-) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^-) (-"h"^2 - 0)/"h"`
= `lim_("h" -> 0^-) - "h"`
= 0
Now Rf ′(0) = `lim_("h" -> 0^+) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^+) ("h"^2 - 0)/"h"`
= `lim_("h" -> 0^+) "h"`
= 0
Since the left hand derivative and right hand derivative both are equal, hence f is differentiable at x = 0.
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
sin (ax + b)
Differentiate the function with respect to x.
`sec(tan (sqrtx))`
Differentiate the function with respect to x.
`(sin (ax + b))/cos (cx + d)`
Differentiate the function with respect to x.
cos x3 . sin2 (x5)
Differentiate the function with respect to x.
`cos (sqrtx)`
Prove that the function f given by f(x) = |x − 1|, x ∈ R is not differentiable at x = 1.
Differentiate the function with respect to x:
sin3 x + cos6 x
Differentiate the function with respect to x:
`sin^(–1)(xsqrtx), 0 ≤ x ≤ 1`
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that `[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.
If y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx = |(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`.
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
|sinx| is a differentiable function for every value of x.
Show that the function f(x) = |sin x + cos x| is continuous at x = π.
`cos(tan sqrt(x + 1))`
`sin^-1 1/sqrt(x + 1)`
(sin x)cosx
sinmx . cosnx
(x + 1)2(x + 2)3(x + 3)4
`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
If k be an integer, then `lim_("x" -> "k") ("x" - ["x"])` ____________.
If `"f"("x") = ("sin" ("e"^("x"-2) - 1))/("log" ("x" - 1)), "x" ne 2 and "f" ("x") = "k"` for x = 2, then value of k for which f is continuous is ____________.
If `y = (x + sqrt(1 + x^2))^n`, then `(1 + x^2) (d^2y)/(dx^2) + x (dy)/(dx)` is
If sin y = x sin (a + y), then value of dy/dx is
If f(x) = `{{:(x^2"," if x ≥ 1),(x"," if x < 1):}`, then show that f is not differentiable at x = 1.
Which expression defines the derivative of a real function \[f\] at a point \[c\] in its domain?
If \[u\] and \[v\] are differentiable functions, what is \[(uv)'\]?
What is \[\frac{d}{dx}(\cos x)\]?
What is \[\frac{d}{dx}(\tan x)\]?
What is the practical condition for differentiability at a point?
Which conclusion establishes that \[f\] is continuous at \[x=c\]?
For \[f(x)=|x|\], what is the left-hand derivative at \[x=0\]?
Why is \[|x|\] not differentiable at \[x=0\]?
