Derivatives are one of the foundational concepts in calculus, essential for understanding rates of change, slopes of curves, and optimization problems. In this blog post, I will provide a comprehensive list of derivative formulas to help you master this important topic.
Basic Derivative Rules
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\bullet \textbf{ Derivative of x: } \frac{d}{dx}(x) = 1 \\
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\bullet \textbf{ Derivative of 1/x: } \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \\
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\bullet \textbf{ Constant Rule: } \frac{d}{dx}[c] = 0 \text{, where c is a constant} \\
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\bullet \textbf{ Power Rule: } \frac{d}{dx}[x^n] = n \cdot x^{n-1} \text{, where n is any real number} \\
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\bullet \textbf{ Constant Multiple Rule: } \frac{d}{dx}[c \cdot g(x)] = c \cdot \frac{d}{dx}[g(x)] \text{, where c is a constant} \\
\displaystyle
\bullet \textbf{ Sum/Difference Rule: } \frac{d}{dx}[g(x) \pm h(x)] = \frac{d}{dx}[g(x)] \pm \frac{d}{dx}[h(x)] \\
\)
Derivatives of Exponential and Logarithmic Functions
\(
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\bullet \textbf{ Exponential Function (Base e): } \frac{d}{dx}[e^x] = e^x \\
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\bullet \textbf{ Exponential Function (Base a): } \frac{d}{dx}[a^x] = a^x \cdot \ln(a) \text{, where a > 0} \\
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\bullet \textbf{ Natural Logarithm Function: } \frac{d}{dx}[\ln(x)] = \frac{1}{x} \\
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\bullet \textbf{ Logarithmic Function (Base a): } \frac{d}{dx}[\log_a(x)] = \frac{1}{x \cdot \ln(a)} \\
\)
Derivatives of Trigonometric Functions
\(
\displaystyle
\bullet \textbf{ Sine Function: } \frac{d}{dx}[\sin(x)] = \cos(x) \\
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\bullet \textbf{ Cosine Function: } \frac{d}{dx}[\cos(x)] = -\sin(x) \\
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\bullet \textbf{ Tangent Function: } \frac{d}{dx}[\tan(x)] = \sec^2(x) \\
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\bullet \textbf{ Cosecant Function: } \frac{d}{dx}[\csc(x)] = -\csc(x) \cot(x) \\
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\bullet \textbf{ Secant Function: } \frac{d}{dx}[\sec(x)] = \sec(x) \tan(x) \\
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\bullet \textbf{ Cotangent Function: } \frac{d}{dx}[\cot(x)] = -\csc^2(x) \\
\)
Derivatives of Inverse Trigonometric Functions
\(
\displaystyle
\bullet \textbf{ Inverse Sine Function: } \frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 – x^2}} \\
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\bullet \textbf{ Inverse Cosine Function: } \frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 – x^2}} \\
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\bullet \textbf{ Inverse Tangent Function: } \frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2} \\
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\bullet \textbf{ Inverse Cosecant Function: } \frac{d}{dx}[\text{arccsc}(x)] = -\frac{1}{|x| \sqrt{x^2 – 1}} \\
\displaystyle
\bullet \textbf{ Inverse Secant Function: } \frac{d}{dx}[\text{arcsec}(x)] = \frac{1}{|x| \sqrt{x^2 – 1}} \\
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\bullet \textbf{ Inverse Cotangent Function: } \frac{d}{dx}[\text{arccot}(x)] = -\frac{1}{1 + x^2} \\
\)
Advanced Derivative Rules
\(
\displaystyle
\bullet \textbf{ Product Rule: } \frac{d}{dx}[g(x) \cdot h(x)] = \frac{d}{dx}[g(x)] \cdot h(x) + g(x) \cdot \frac{d}{dx}[h(x)] \\
\displaystyle
\bullet \textbf{ Quotient Rule: } \frac{d}{dx}\left[\frac{g(x)}{h(x)}\right] = \frac{\frac{d}{dx}[g(x)] \cdot h(x) – g(x) \cdot \frac{d}{dx}[h(x)]}{[h(x)]^2} \\
\displaystyle
\bullet \textbf{ Chain Rule: } \frac{d}{dx}[g(h(x))] = \frac{d}{dx}g((h(x))) \cdot \frac{d}{dx}[h(x)] \\
\)
