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If $ f(x) = 5^x $, show that

$ \dfrac{f (x + h) - f(x)}{h} = 5^x \left(\frac{5^h - 1}{h}\right) $

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$5^{x}\left(\frac{5^{h}-1}{h}\right)$

01:09

Jeffrey Payo

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 4

Exponential Functions

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Campbell University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:31

A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Okay, so we have this expression that we want to evaluate for F of X equals five to the X power. So let's go ahead and substitute five to the X in for f of X and five to the X plus H in four f of X plus h sweet five to the X plus H minus five to the X over H. Now we want to simplify that. So remember that when you see five to the X plus H power, it's equivalent to five to the X power power times five to the age power. Think about your exponents rules. And if you were given five to the X times five to the age, you would add the exponents. And now this is still all over H. Okay, so notice that what we can do is factor five to the X power out of both terms in the numerator. And now we have five to the X power multiplied by pi to the H power minus one over H. And that's exactly what we're asked to show

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