Find dy/dx if $y = (ln(x))^(cos(x))$.

Step 1: Take logarithm on both sides

To differentiate a function of the form f(x)^{g(x)}, we use logarithmic differentiation. We start by taking the natural logarithm on both sides of the equation.

Step 2: Apply logarithm property

Using the logarithm property $\ln(a^b) = b \ln a$ we can simplify the right-hand side of the equation.

Step 3: Differentiate both sides with respect to x

Now, we differentiate both sides of the equation with respect to x. On the left side, we use the chain rule. On the right side, we will use the product rule.

Step 4: Apply product rule and chain rule

Applying the product rule $\frac{d}{dx}(uv) = u'v + uv'$ to the right side, where $u = \cos x$ and $v = \ln(\ln x)$. We also use the chain rule for differentiating $\ln(\ln x)$.

Step 5: Isolate dy/dx

To find $\frac{dy}{dx}$, we multiply both sides of the equation by y.

Step 6: Substitute y back into the equation

Finally, we substitute the original expression for y, which is ($\ln x$)^{$\cos x$}, back into the equation to get the derivative in terms of x only.

More from this chapter

• 2. Consider the function f(x) = 3cos(e^(5x-3)) + (4)/(x) Find f'(x) [(d)/(dx)(cos x) = -sin x]
• **प्रश्नावली 5.2** प्रश्न 1 से 8 में x के सापेक्ष निम्नलिखित फलनों का अवकलन कीजिए: 1. sin(x² + 5) 2. cos(sin x) 3. sin(ax + b) 4. sec(tan(√x)) 5. (sin(ax + b))/
• 1. Prove that the function f(x) = 2x² - 3x + 2 is continuous at x = 1. 2. The function f(x) is defined as follows: f(x) = 2x - 3, & x < 2 x - 1, & x ≥ 2 Prove t
• Differentiate the following w.r.t. x. (i) cos^-1(sin x) (ii) tan^-1(sin x / (1 + cos x)) (iii) sin^-1(2^(x+1) / (1 + 4^x))
• If x = t² and y = t³, then (d² y)/(dx²) is equal to:
• Find (d y)/(d x), if x=a(θ+sin θ), y=a(1-cos θ).
• Differentiate the following w.r.t. x. (i) cos ^(-1)(sin x) (ii) tan ^(-1)((sin x)/(1+cos x)) (iii) sin ^(-1)((2^(x+1))/(1+4^x))
• Let the function f(x)=(x²+1)|x²-ax+2|+cos|x| be not differentiable at the two points x=α=2 and x=β. Then the distance of the point (α, β) from the line 12x+5y+1

All Continuity and Differentiability solutions →