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Byju's Answer

Standard XII

Mathematics

Monotonically Increasing Functions

Say true or f...

Question

Say true or false.
The derivative of real-valued function $f (x)$ at $a$ is defined by, $lim h \to 0 \frac{f (a + h) - f (a)}{h}$

True

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False

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Solution

The correct option is A True
The derivative of a function $f (x)$ at a point $a$ is ${lim}_{h \to 0} \frac{f (a + h) - f (a)}{h}$ .
Therefore, the given statement is true.

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Similar questions

State whether the given statement is True or False.

The derivative of a function is defined at

a

, provided

l i m_{h \to 0} \frac{f (a + h) - f (a)}{h}

exists.

Left hand derivative and right hand derivative of a function

f (x)

at a point

x = a

are defined as

f^{'} (a^{-}) = lim h \to 0^{+} \frac{f (a) - f (a - h)}{h} = lim h \to 0^{-} \frac{f (a) - f (a - h)}{h} = lim x \to a^{+} \frac{f (a) - f (x)}{a - x}

respectively

Let

f

be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

The statement

lim h \to 0 \frac{f (- x) - f (- x - h)}{h} = lim h \to 0 \frac{f (x) - f (x - h)}{- h}

implies that for all x

ϵ R

Left hand derivative and right hand derivative of a function

f (x)

at a point

x = a

are defined as

f^{'} (a^{-}) = lim h \to 0^{+} \frac{f (a) - f (a - h)}{h} = lim h \to 0^{-} \frac{f (a) - f (a - h)}{h} = lim x \to a^{+} \frac{f (a) - f (x)}{a - x}

respectively

Let

f

be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

f

is even, which of the following is Right hand derivative of

f^{'}

x = a

Left hand derivative and right hand derivative of a function

f (x)

at a point

x = a

are defined as

f^{'} (a^{-}) = lim h \to 0^{+} \frac{f (a) - f (a - h)}{h} = lim h \to 0^{-} \frac{f (a) - f (a - h)}{h} = lim x \to a^{+} \frac{f (a) - f (x)}{a - x}

respectively

Let

f

be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

f

is odd, which of the following is Left-hand derivative of

f

x = a

Q. Assertion :Derivative of

3 cot x + 5 cosec x

- cosec x (3 cosec x + 5 cot x)

Reason:

f^{'} (a) = lim h \to 0 \frac{f (a + h) - f (a)}{h}

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Say true or false.The derivative of real-valued function f(x) at a is defined by, limh→0f(a+h)−f(a)h

The correct option is A TrueThe derivative of a function f(x) at a point a is limh→0f(a+h)−f(a)h.Therefore, the given statement is true.

Say true or false.
The derivative of real-valued function $f (x)$ at $a$ is defined by, $lim h \to 0 \frac{f (a + h) - f (a)}{h}$

The correct option is A True
The derivative of a function $f (x)$ at a point $a$ is ${lim}_{h \to 0} \frac{f (a + h) - f (a)}{h}$ .
Therefore, the given statement is true.