Lectures>Practice

# Exercises

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## Exercises

Find the derivatives of the following functions.

 $f(x) = \ln(x^2 + 1)$ $e^{xy+1}=y$ $g(x) = \log_3(x^2 + 1)$ $e^{\ln(x+y+1)} = x$ $f(x) = e^{x^2 + 1}$ $\ln(x+y)=x+y+1$ $g(x) = 3^{x^2 + 1}$ $h(x) = x\ln(3x+1)$ $f(x) = 4\log(e^x+1)$ $f(x) = e^{\cos x}$ $f(x) = e^{(x+1)^2}$ $f(x) = x^2 + 4^{2x-1}$ $g(x) = \ln\ln x$ $m(x) = \cos(\ln x)$ $t(x) = \frac{e^x}{e^x+1}$ $f(x) = \frac{\ln x}{x + 1}$ $g(x) = \ln(\sqrt{x+1} + x)$ $m(x) = \ln \frac{x+a}{x-a}$
1. Find a formula for the nth derivative of $f(x) = \ln x$.
2. Find a formula for the nth derivative of $g(x) = e^{2x}$.
3. Find a formula for the nth derivative of $h(x) = xe^{x}$.

## Explorations

Logarithmic differentiation is a method where you start by taking the logarithm of a function, simplify it using logarithm properties then differentiate it implicitly. Thie method is particularly useful in situations where there are variables both in the base of an expression and in its exponent. Use this procedure to differentiate the following functions.

 $y = x^x$ $y = x^{\cos x}$ $y = (\sqrt{x})^{x}$ $y = \sqrt{x}$ $y = (x^2 + 1)^2(x^3 - 1)^4$
 $y = x^x$

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