Find the derivative of:
1. Xlog base e (log base e X)
2. log base x 7
3. (sinh x)^2 - (cosh x)^2
do you mean
y = x ln(ln x)?
if so, then
y' = ln(lnx)) + x * 1/lnx * 1/x
= ln(lnx) + 1/lnx
y = log_x(7) = ln7/lnx
if u = lnx, then
y = ln7 u^-1
y' = -ln7 u^-2 u'
...
since cosh^2 = 1+sinh^2,
sinh^2-cosh^2 = -1
so the derivative is zero
To find the derivatives of the given functions, we can use different rules and properties of derivatives. Let's solve them step by step.
1. To find the derivative of X*log base e (log base e X), we can apply the chain rule.
Let's denote the given function as f(x) = X * log_e(log_e(X)).
Step 1: Apply the chain rule.
- Differentiate the outer function (X) with respect to x, which gives us 1.
- Multiply it by the derivative of the inner function (log_e(log_e(X))).
Step 2: Differentiate the inner function.
- Using logarithmic differentiation, we can rewrite log_e(log_e(X)) as (ln(log_e(X))) / (ln(e)).
- Differentiate (ln(log_e(X))), which means differentiating the logarithm of the inner function.
- Apply the chain rule again.
Let's denote the inner function as g(X) = log_e(X).
Step 2.1: Differentiate the inner function.
- Using the logarithmic differentiation, we can rewrite log_e(X) as (ln(X)) / (ln(e)).
- Differentiate (ln(X)), which gives us (1 / X) * (1 / ln(e)).
- Simplify it to 1 / X.
Step 2.2: Apply the chain rule.
- Multiply the derivative of g(X) by the derivative of the inner function, which gives us (1 / X) * (1 / ln(e)).
Step 3: Combine the results.
- Multiply the derivative of the outer function (1) with the derivative of the inner function [(1 / X) * (1 / ln(e))].
- Simplify it to 1 / (X * ln(e) * ln(e)).
Therefore, the derivative of X*log base e (log base e X) is 1 / (X * ln(e) * ln(e)), or simply 1 / (X * ln^2(e)).
2. To find the derivative of log base x 7, we can use the logarithmic differentiation method and the chain rule.
Let's denote the given function as f(x) = log_x(7).
Step 1: Rewrite the function using logarithmic properties.
- log_x(7) can be rewritten as (ln(7)) / (ln(x)).
Step 2: Differentiate the function using logarithmic differentiation.
- Differentiate both parts of the function with respect to x.
- Apply the chain rule while differentiating ln(x).
Step 2.1: Differentiate the numerator.
- Differentiate ln(7), which gives us 0 (since 7 is a constant).
Step 2.2: Differentiate the denominator.
- Differentiate ln(x) using the chain rule.
- The derivative of ln(x) is (1 / x).
Step 3: Combine the results.
- Divide the derivative of the numerator (0) by the denominator (ln(x)), which gives us 0.
- Therefore, the derivative of log base x 7 is 0.
3. To find the derivative of (sinh(x))^2 - (cosh(x))^2, we can use the power rule and the derivatives of hyperbolic sine and cosine functions.
Let's denote the given function as f(x) = (sinh(x))^2 - (cosh(x))^2.
Step 1: Differentiate each term of the function separately.
- Differentiate (sinh(x))^2 using the power rule.
- The derivative of (sinh(x))^2 is 2 * sinh(x) * cosh(x).
- Differentiate (cosh(x))^2 using the power rule.
- The derivative of (cosh(x))^2 is 2 * cosh(x) * sinh(x).
Step 2: Combine the results.
- The derivative of (sinh(x))^2 - (cosh(x))^2 is (2 * sinh(x) * cosh(x)) - (2 * cosh(x) * sinh(x)).
- By simplifying, we get 0.
Therefore, the derivative of (sinh(x))^2 - (cosh(x))^2 is 0.