Mathematical Expressions for Rate of Change or Slope Computation
In the realm of calculus, derivatives play a crucial role in understanding a function's behaviour and rate of change. Here, we delve into the differentiation of exponential and logarithmic functions, going beyond the basic rules.
Firstly, let's recall the derivative of some common functions. The derivative of the hyperbolic function is , while the derivative of the logarithmic function is . The derivative of a constant function is, naturally, zero.
When it comes to exponential and logarithmic functions, the chain rule, logarithmic differentiation, and recognising the inverse relationship become essential tools.
The chain rule helps in handling inner functions within exponential or logarithmic functions. For an exponential function with a general base and function in the exponent, the formula is:
[ \frac{d}{dx} \left(a^{g(x)}\right) = a^{g(x)} \cdot \ln(a) \cdot g'(x) ]
For the natural exponential function with a function in the exponent, the formula simplifies to:
[ \frac{d}{dx} \left(e^{g(x)}\right) = e^{g(x)} \cdot g'(x) ]
Logarithmic differentiation is a technique used when differentiating complicated functions that involve products, quotients, or variable exponents. The steps are:
- Take the natural logarithm of both sides.
- Simplify using log properties (product, quotient, power rules).
- Differentiate implicitly.
- Solve for the derivative.
For example, if , then taking logs and differentiating yields:
[ \frac{dy}{dx} = y \left( v'(x) \ln(u(x)) + v(x) \frac{u'(x)}{u(x)} \right) ]
Logarithmic differentiation simplifies the differentiation of complex exponentials.
The exponential function also satisfies the key property:
[ \exp(x + y) = \exp(x) \cdot \exp(y) ]
which underlies many properties and can help in differentiating sums inside exponents.
The second derivative, third derivative, and so on, can be found by repeatedly differentiating the second derivative with respect to x. For instance, the second derivative of the function is .
In summary, the differentiation of exponential and logarithmic functions beyond the basics makes use of the chain rule to handle inner functions, logarithmic differentiation to simplify complicated expressions, recognition of the inverse relationship to simplify problems, and the properties of logs and exponents to rewrite functions before differentiation. These tools extend basic formulas like and to much more complex, composite functions.
Some derivatives not provided in the example include the third derivative of the function , the second derivative of the function , and the third derivative of the function . The derivative of the inverse trigonometric function is . The derivative of the power function is . The second derivative of the function is not provided. The nth derivative of a function can be found by repeatedly differentiating the (n-1)th derivative with respect to x.
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