Advertisements
Advertisements
Question
(sin x)cosx
Advertisements
Solution
Let y = (sin x)cosx
⇒ log y = log(sin x)cosx = cosx log(sinx)
Differentiate both sides w.r.t. x, we get
⇒ `1/y * "dy"/"dx" = cos x * "d"/"dx" (log sin x) + log sin x "d"/"dx" (cos x)`
= `cos x * 1/sinx * "d"/"dx" (sin x) + log sin x * (- sin x)`
= `cot x * cos x - log (sin x) * sin x`
∴ `"dy"/"dx" = y[cot x cos x - sin x * log(sin x)]`
= (sin x)cosx [cot x cos x – sin x · log(sin x)]
APPEARS IN
RELATED QUESTIONS
Differentiate the function with respect to x.
cos (sin x)
Differentiate the function with respect to x.
cos x3 . sin2 (x5)
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:
`(cos^(-1) x/2)/sqrt(2x+7)`, −2 < x < 2
If f(x) = |x|3, show that f"(x) exists for all real x and find it.
`"If y" = (sec^-1 "x")^2 , "x" > 0 "show that" "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`
If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.
|sinx| is a differentiable function for every value of x.
sinn (ax2 + bx + c)
sinx2 + sin2x + sin2(x2)
(x + 1)2(x + 2)3(x + 3)4
`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`
`tan^-1 (("a"cosx - "b"sinx)/("b"cosx - "a"sinx)), - pi/2 < x < pi/2` and `"a"/"b" tan x > -1`
`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`
If xm . yn = (x + y)m+n, prove that `("d"^2"y")/("dx"^2)` = 0
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
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 ____________.
A function is said to be continuous for x ∈ R, if ____________.
`d/(dx)[sin^-1(xsqrt(1 - x) - sqrt(x)sqrt(1 - x^2))]` is equal to
Let f: R→R and f be a differentiable function such that f(x + 2y) = f(x) + 4f(y) + 2y(2x – 1) ∀ x, y ∈ R and f’(0) = 1, then f(3) + f’(3) is ______.
If f(x) = `{{:((sin(p + 1)x + sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x + x^2) - sqrt(x))/(x^(3//2)),",", x > 0):}`
is continuous at x = 0, then the ordered pair (p, q) is equal to ______.
If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.
If f(x) = | cos x |, then `f((3π)/4)` is ______.
The set of all points where the function f(x) = x + |x| is differentiable, is ______.
Which expression defines the derivative of a real function \[f\] at a point \[c\] in its domain?
What is \[\frac{d}{dx}(\cos x)\]?
Which limit is the right-hand derivative at \[x=c\]?
When is a function differentiable on an open interval \[(a,b)\]?
What does differentiability at a point mean?
