Use logarithmic differentiation to find the derivative of the function.
y = (cos 2x)x
y = (cos 2x)^x
log y = x log(cos 2x)
1/y y' = log(cos 2x) + x/cos2x (-2sin2x)
y' = (cos2x)^x (log cos2x - 2x*tan2x)
It is interesting to note that if
y = u^v
y' = vu^(v-1) u' + (logv)(u^v)(v')
If v is a constant, v'=0, so we have the familiar power rule.
If u is a constant, u'=0 and we have the familiar exponentiation rule
Sure, let's have some logarithmic fun!
To find the derivative of the function y = (cos 2x)x, we'll need to use logarithmic differentiation.
Step 1: Take the natural logarithm (ln) of both sides of the equation:
ln(y) = ln[(cos 2x)x]
Step 2: Apply the logarithmic properties to simplify the expression. We can use the laws of logarithms to break down the function into more manageable parts.
ln(y) = x ln(cos 2x)
Step 3: Differentiate both sides of the equation with respect to x. Remember that when we differentiate ln(y), we also need to use the chain rule.
(1/y) * y' = ln(cos 2x) + x (-sin 2x)(2)
Step 4: Simplify the expression by combining like terms and solving for y'.
y' = y * [ln(cos 2x) - 2x sin 2x]
Now we can substitute our original function back in:
y' = (cos 2x)x * [ln(cos 2x) - 2x sin 2x]
And that's it! We've found the derivative of the function using logarithmic differentiation. Remember, it's a logarithmic party in the world of derivatives!
To find the derivative of the function y = (cos 2x)x using logarithmic differentiation, follow these steps:
Step 1: Take the natural logarithm of both sides of the equation: ln(y) = ln((cos 2x)x).
Step 2: Apply the logarithmic property of ln(a^b) = b * ln(a) to the right side of the equation: ln(y) = x * ln((cos 2x)).
Step 3: Differentiate both sides of the equation implicitly with respect to x. Remember that when differentiating ln(y), you need to use the chain rule.
On the left side, the derivative of ln(y) with respect to x is (1 / y) * y'. On the right side, you have x * [the derivative of ln((cos 2x)) with respect to x] + ln((cos 2x)) * [the derivative of x with respect to x], which simplifies to x * [the derivative of ln((cos 2x)) with respect to x] + ln((cos 2x)).
Step 4: Simplify the right side of the equation. The derivative of ln((cos 2x)) with respect to x can be found using the chain rule.
Let u = cos 2x, then the derivative of ln(u) with respect to x is (1 / u) * u', where u' is the derivative of u with respect to x.
Taking the derivative of cos 2x with respect to x, we get u' = -sin 2x * 2 = -2sin 2x.
Therefore, the derivative of ln((cos 2x)) with respect to x is (1 / cos 2x) * (-2sin 2x) = -2sin 2x / cos 2x.
Step 5: Substitute the value for the derivative of ln((cos 2x)) into the equation: (1 / y) * y' = x * (-2sin 2x / cos 2x) + ln((cos 2x)).
Step 6: Solve for y' (the derivative of y) by multiplying both sides of the equation by y: y' = y * [x * (-2sin 2x / cos 2x) + ln((cos 2x))].
Step 7: Substitute the value of y = (cos 2x)x back into the equation: y' = [(cos 2x)x] * [x * (-2sin 2x / cos 2x) + ln((cos 2x))].
Simplifying further, we get: y' = x * (-2sin 2x) + x * ln((cos 2x)).
To find the derivative of the function y = (cos 2x)x using logarithmic differentiation, follow these steps:
Step 1: Take the natural logarithm (ln) of both sides of the equation:
ln(y) = ln[(cos 2x)x]
Step 2: Use the properties of logarithms to simplify the expression:
ln(y) = x ln(cos 2x)
Step 3: Differentiate both sides of the equation implicitly with respect to x:
1/y * y' = ln(cos 2x) + x * d/dx[ln(cos 2x)]
Step 4: Use the chain rule to find the derivative of ln(cos 2x):
d/dx[ln(cos 2x)] = -2 sin 2x / cos 2x
Step 5: Substitute the derivative back into the equation:
1/y * y' = ln(cos 2x) - 2x sin 2x / cos 2x
Step 6: Multiply both sides of the equation by y to solve for y':
y' = y * (ln(cos 2x) - 2x sin 2x / cos 2x)
Step 7: Substitute the original function y = (cos 2x)x back into the equation for y':
y' = (cos 2x)x * (ln(cos 2x) - 2x sin 2x / cos 2x)
Therefore, the derivative of the function y = (cos 2x)x using logarithmic differentiation is y' = (cos 2x)x * (ln(cos 2x) - 2x sin 2x / cos 2x).