### Continuity and Differentiability Exercise 04

## Question:

Differentiate the given function w.r.t. x. log(logx),x>1

## Answer:

Step 1: Use the chain rule,

d/dx(log(logx)) = (1/logx) * (1/x)

Step 2: Simplify,

d/dx(log(logx)) = (1/xlogx)

## Question:

Differentiate the given function w.r.t. x. cos(logx+e^x),x>0

## Answer:

Step 1: Use Chain Rule to differentiate the given function.

Step 2: Differentiate logx with respect to x.

Step 3: Differentiate e^x with respect to x.

Step 4: Multiply the derivatives from Steps 2 and 3 and add them together.

Step 5: Differentiate cos(logx+e^x) with respect to x using the result from Step 4.

Answer: -sin(logx+e^x)*(1/x + e^x)

## Question:

Differentiate the given function w.r.t. x. esin^−1x

## Answer:

Step 1: Rewrite the given function in terms of y, where y = sin^−1x

Step 2: Differentiate the function w.r.t. y

Step 3: Use the chain rule to differentiate the function w.r.t. x

Differentiate w.r.t. y: e^y

Using the chain rule: e^y * (1/√(1-x^2))

## Question:

Differentiate the given function w.r.t. x. e^x^3

## Answer:

Step 1: Rewrite the function as (e^x)^3

Step 2: Use the Chain Rule: (e^x)^3 = (3*e^x)*(e^x)^2

Step 3: Differentiate each term: (3*e^x)’ = 3*e^x; (e^x)^2 = 2*e^x*(e^x)'

Step 4: Combine the terms: (3*e^x)’ + (e^x)^2 = 3*e^x + 2*e^x*(e^x)'

Step 5: Differentiate the last term: 2*e^x*(e^x)’ = 2*e^x*e^x = 2*e^2x

Step 6: Combine the terms: 3*e^x + 2*e^2x = 3*e^x + 2*e^2x

Answer: The derivative of e^x^3 w.r.t. x is 3*e^x + 2*e^2x.

## Question:

Differentiate the given function w.r.t. x. y=log(cose^x)

## Answer:

Step 1: Take the natural logarithm of both sides of the equation.

log(y) = log(cose^x)

Step 2: Apply the chain rule on the right side of the equation.

log(y) = x*log(cose)

Step 3: Differentiate both sides of the equation with respect to x.

(d/dx)log(y) = (d/dx)(x*log(cose))

Step 4: Simplify the right side of the equation.

(d/dx)log(y) = log(cose) + x*(d/dx)(log(cose))

Step 5: Apply the chain rule on the right side of the equation.

(d/dx)log(y) = log(cose) + x*(1/cose)*(d/dx)(cose)

Step 6: Simplify the right side of the equation.

(d/dx)log(y) = log(cose) - x*(1/cose^2)

Step 7: The final answer is:

(d/dx)log(y) = log(cose) - x*(1/cose^2)

## Question:

Differentiate the given function w.r.t. x. y=e^x+e^x^2+…+e^x^5

## Answer:

Step 1: Rewrite the given function as a summation of powers of e^x as follows: y= e^x + e^2x + e^3x + e^4x + e^5x

Step 2: Use the power rule to differentiate the function w.r.t. x as follows:
dy/dx = (1*e^x) + (2*e^2x) + (3*e^3x) + (4*e^4x) + (5*e^5x)

Step 3: Simplify the expression to get the final answer as follows: dy/dx = e^x + 2e^2x + 3e^3x + 4e^4x + 5e^5x

## Question:

Differentiate the given function w.r.t. x. y=sin(tan^−1e^−x)

## Answer:

Step 1: Use the chain rule to differentiate the function.

Step 2: Differentiate the inner function tan^−1e^−x with respect to x.

Step 3: Differentiate the outer function sin(tan^−1e^−x) with respect to x.

Step 4: Combine the two derivatives to get the final answer.

Answer: -e^−xcos(tan^−1e^−x)

## Question:

Differentiate the given function w.r.t. x. cosx/logx,x>0

## Answer:

Given: f(x) = cosx/logx, x > 0

Step 1: Rewrite the function as f(x) = (cosx) (1/logx)

Step 2: Apply the product rule of differentiation to f(x):

f’(x) = (cosx) (1/logx)’ + (1/logx) (cosx)'

Step 3: Differentiate the first term (cosx) (1/logx)':

(cosx) (1/logx)’ = (cosx) (-1/logx^2)

Step 4: Differentiate the second term (1/logx) (cosx)':

(1/logx) (cosx)’ = (-1/logx^2) (cosx)'

Step 5: Differentiate the third term (cosx)':

(cosx)’ = -sinx

Step 6: Substitute the derivatives of the three terms into the equation for f’(x):

f’(x) = (cosx) (-1/logx^2) + (-1/logx^2) (-sinx)

Step 7: Simplify the equation:

f’(x) = -(cosx + sinx)/logx^2

## Question:

Differentiate the given function w.r.t. x. y=√e^√x,x>0

## Answer:

Step 1: Rewrite y in terms of exponential form: y = e^(1/2√x)

Step 2: Take the derivative of both sides with respect to x: dy/dx = (1/2)e^(1/2√x) × (1/2)x^(-1/2)

Step 3: Simplify the expression: dy/dx = (1/4)e^(1/2√x) x^(-1/2)

## Question:

Differentiate the given function w.r.t. x. e^x/sinx

## Answer:

Step 1: Rewrite the given function as a product of two functions: e^x and 1/sinx

Step 2: Use the product rule to differentiate the given function:

(e^x)(d/dx (1/sinx)) + (1/sinx)(d/dx (e^x))

Step 3: Differentiate the first term:

(e^x)(-cosx/sinx^2)

Step 4: Differentiate the second term:

(1/sinx)(e^x)

Step 5: Combine the two terms:

(e^x)(-cosx/sinx^2) + (1/sinx)(e^x)

Step 6: Simplify the expression:

e^x(-cosx/sinx^2 + 1/sinx)