You CAN Ace Calculus

 basic derivative rules power rule product rule quotient rule For the basic exponential derivatives you do not need the chain rule. But we discuss it on this page. Each section is labeled. So if you have not studied the chain rule yet, you can read the sections that apply to you and then come back here once you have studied it.

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > derivatives > exponential derivatives

Exponential Derivative

This is one of the easiest rules you will learn.

Basic Exponential Rule

$$\displaystyle{ \frac{d}{dt}[e^t] = e^t }$$

Exponential With Chain Rule

$$\displaystyle{ \frac{d}{dt}[e^u] = e^u \frac{du}{dt} }$$

It looks like we didn't do anything here. However, the exponential function is the only function whose derivative is itself.
Before we go on, let's watch a video that gives an intuitive explanation of the derivative of exponential functions and why $$f(x)=e^x$$ is special.

### 3Blue1Brown - Derivatives of exponentials [13min-49secs]

video by 3Blue1Brown

Okay, so what do you do if you have a base other than $$e$$? The formula is fairly straightforward but let's derive from our rules of logarithms. Since we don't like to memorize formulas and we already know the logarithm rules, why not just derive it when we need it, since we don't use it very much?

So, let's convert $$y=a^x$$, where $$a$$ is a constant, into a form with $$e$$.
$$\begin{array}{rcl} y & = & a^x \\ \ln(y) & = & \ln(a^x) \\ & = & x\ln(a) \\ e^{\ln(y)} & = & e^{x\ln(a)} \\ y & = & e^{x\ln(a)} \end{array}$$

So now when we take the derivative of $$y = a^x$$, we can actually take the derivative of $$y=e^{x\ln(a)}$$. Using the chain rule, we have $$(a^x)' = (\ln(a))e^{x\ln(a)}$$. Notice how didn't have to memorize this formula. We used the logarithm rules we already know.

This next video goes through all the explanation again. It is always good to get explanations from different sources since it will help you understand the material better.

### PatrickJMT - Derivatives of Exponential Functions [5min-49secs]

video by PatrickJMT

Here is an interesting video that shows how to get the equation for the derivative of $$f(x)=a^x$$ another way. He shows that $$\displaystyle{f'(x)=\frac{d[a^x]}{dx}=a^x f'(0)}$$. This is an interesting and unusual way to think about the derivative.

### Dr Chris Tisdell - Derivative of exponentials [9min-15secs]

video by Dr Chris Tisdell

So far, we've only been looking at equations with exponential functions. Here is a video discussing the graph, the derivative and the tangent line of three exponential functions. This helps you get more of an intuitive feel for this function and it's derivative.

video by MathTV

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Exponential Derivatives - Practice Problems Conversion

[1-1066] - [2-1067] - [3-1068] - [4-1069] - [5-1074] - [6-1076] - [7-973] - [8-1075] - [9-1077]

[10-1078] - [11-986] - [12-1079]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

Unless otherwise instructed, calculate the derivative of these functions. Here are a few practice problems that do not require the chain rule.

$$y=e^x(x+x\sqrt{x})$$

Problem Statement

Calculate the derivative of $$y=e^x(x+x\sqrt{x})$$.

Solution

### 1069 solution video

video by Krista King Math

$$f(x)=4^x+3e^x+x^4$$

Problem Statement

Calculate the derivative of $$f(x)=4^x+3e^x+x^4$$.

Solution

### 1074 solution video

video by PatrickJMT

$$f(x)=e^xx^2$$

Problem Statement

Calculate the derivative of $$f(x)=e^xx^2$$.

Solution

### 1076 solution video

video by PatrickJMT

These practice problems require the chain rule. As before calculate the derivative of these functions, unless otherwise instructed.

Basic Problems

$$f(x)=(x^2-1)e^{-x}$$

Problem Statement

Calculate the derivative of $$f(x)=(x^2-1)e^{-x}$$.

Solution

### 1066 solution video

video by Krista King Math

$$\displaystyle{f(x)=xe^{\sqrt{x}}}$$

Problem Statement

Calculate the derivative of $$\displaystyle{f(x)=xe^{\sqrt{x}}}$$ .

Solution

### 1067 solution video

video by Krista King Math

$$\displaystyle{f(x)=\frac{1-e^{-x}}{x}}$$

Problem Statement

Calculate the derivative of $$\displaystyle{f(x)=\frac{1-e^{-x}}{x}}$$.

Solution

### 1068 solution video

video by Krista King Math

$$\displaystyle{ 3e^{ x^2+7 } }$$

Problem Statement

Use the chain rule to calculate the derivative of $$\displaystyle{ 3e^{ x^2+7 } }$$.

$$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }$$

Problem Statement

Use the chain rule to calculate the derivative of $$\displaystyle{ 3e^{ x^2+7 } }$$.

Solution

 $$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }$$ $$\displaystyle{ 3e^{x^2+7} \cdot \frac{d}{dx}[x^2+7] }$$ $$3e^{x^2+7} \cdot (2x)$$ $$6xe^{x^2+7}$$

Another way to work this is with the substitution method.

 let $$u=x^2+7$$ $$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }$$ $$\displaystyle{ 3\frac{d[e^u]}{du} \cdot \frac{d[x^2+7]}{dx} }$$ $$3e^u \cdot (2x)$$ $$6xe^{x^2+7}$$

$$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }$$

Intermediate Problems

$$\displaystyle{f(x)=e^{x\sin(2x)}}$$

Problem Statement

Calculate the derivative of $$\displaystyle{f(x)=e^{x\sin(2x)}}$$.

Solution

### 1077 solution video

video by PatrickJMT

$$\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}$$

Problem Statement

Calculate the derivative of $$\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}$$.

Solution

### 1075 solution video

video by PatrickJMT

$$\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}$$

Problem Statement

Calculate the derivative of $$\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}$$.

Solution

### 1078 solution video

video by PatrickJMT

For what values of x does $$h(x)=5e^{5x}-25x$$ have negative derivatives?

Problem Statement

For what values of x does $$h(x)=5e^{5x}-25x$$ have negative derivatives?

Solution

### 1079 solution video

video by PatrickJMT

$$\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }$$

Problem Statement

Use the chain rule to calculate the derivative of $$\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }$$.

Solution

### 986 solution video

video by Krista King Math