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Derivative of $$$- a l m x$$$ with respect to $$$a$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{da} \left(- a l m x\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{da} \left(c f{\left(a \right)}\right) = c \frac{d}{da} \left(f{\left(a \right)}\right)$$$ with $$$c = - l m x$$$ and $$$f{\left(a \right)} = a$$$:
$${\color{red}\left(\frac{d}{da} \left(- a l m x\right)\right)} = {\color{red}\left(- l m x \frac{d}{da} \left(a\right)\right)}$$Apply the power rule $$$\frac{d}{da} \left(a^{n}\right) = n a^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{da} \left(a\right) = 1$$$:
$$- l m x {\color{red}\left(\frac{d}{da} \left(a\right)\right)} = - l m x {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{da} \left(- a l m x\right) = - l m x$$$.
Answer
$$$\frac{d}{da} \left(- a l m x\right) = - l m x$$$A